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ARITHMETIC, 

IN     TWO     PARTS 


PART    FIRST, 

ADVANCED  LESSONS  IN  MENTAL   ARITHMETIC. 

PART    SECOND, 

RULES  AND  EXAMPLES  FOR  PRACTICE  IN 
WRITTEN  ARITHMETIC. 


FOR    COMMON    AND    HIOH    SCHOOLS. 


BY  FREDERIC  A.  ADAMS, 

PRINCIPAL    OF    DUMMER    ACADEMY. 


LOWELL: 
PUBLISHED  BY  D.  BIX-BY  &  CO. 

Boston  :  B.  B.  Mussoy  &  Co. ;  W.  J.  Reynolds  &  Co.    New  Yobk  :  D.  Appleton  &  Co 
PmLADELPHiA  :   Thomas,  Cowperthwait  &  Co.     Baltimore  :  *Cushing  &  Brother. 
Richmond:  Nash"  &  Woodhouse.     Charleston :   McCarter  &  Allen.     Mo- 
bile :  J.  Dobler.    New  Orleans  :  J.'  B.  Steele.     St.  Louis  :  S.  B.  Meech. 
Louisville  :  Morton  &  Griswold.     Cincinnati  :  Derby,  Bradley  &  Co. 
Detroit  :   C.  Morse.     Chicago  :   A.  H.   &  C.  Burley.     Provi- 
DENCB :  C.  Burnet,  Jr.    Portland  :  Hyde,  Lord  ^  Duren. 

- 1848. 


■^■.K 


/,&-69f 


U__l 


Entered  according  to  Act  of  Congress,  in  the  year  1848, 

By  DANIEL  BEXBY, 

In  the  Clerk*8  Office  ocT  the  District  Court  of  the  District  of  Massachusettt. 


aTBHBOTTPED  AND  PEINTKB  BY  DIOKINHOK  &  CO.,  52  WASUIKGTON  ST.,  BOaTOBT. 


PREFACE. 


The  book  here  offered  to  Schools  and  Academies,  had  ite  origin 
in  the  urgent  want  the  author  has  found,  in  the  case  of  his  own 
pupils,  of  a  higher  work  on  Mental  Arithmetic.  Such  a  work, 
he  has  thought,  should  be  constructed  with  reference  to  several 
important  objects. 

It  should  habituate  the  pupil  to  perform,  with  ease  and  readiness, 
mental  operations  upon  somewhat  large  numbers. 

It  should  present  these  operations  in  their  natural  form,  freed 
from  the  inverted  and  mechanical  methods  which  belong  of  neces- 
sity to  operations  in  written  Arithmetic. 

It  should  train  the  student  to  such  a  power  in  apprehending 
the  relations  of  numbers,  as  shall  give  him  an  insight  into  the 
grounds  of  the  rules  of  Arithmetic ;  and,  consequently,  shall  re- 
lease him  from  dependence  on  those  rules;  and  it  should  free 
him  from  the  liability  to  those  wide  mistakes  often  made  in  written 
Arithmetic,  which  appear  so  absurd,  and  are  yet  too  frequent  to 
excite  the  teacher's  surprise. 

A  higher  training  in  Mental  Arithmetic  would  also,  it  is  be- 
lieved, prepare  the  members  of  our  schools,  when  they  should  leave 
their  studies  and  engage  in  the  active  pursuits  of  life,  to  solve 
mentally,  and  with  ease  and  delight,  a  large  share  of  those  ques- 
tions,* of  business  or  curiosity,  for  which  a  process  of  ciphering  is 
^  wrdi^arily  thought  indispensable. 


iv  PREFACE. 

The  study  of  Arithmetic  in  the  schools  of  this  country  received 
its  best  impulse,  unquestionably,  in  the  publication  of  "  Colburn's 
First  Lessons."  So  completely  has  this  little  book  performed  the 
work  within  its  prescribed  sphere,  that  there  is  little  reason  to 
desire  a  change  in  that  particular,  or  to  expect  that  the  work  will,  • 
for  the  present,  be  superseded.  ^Vhoever  would  now  write  a  book 
of  First  Lessons  in  Arithmetic,  must,  it  is  believed,  if  he  would 
write  a  good  one,  walk  the  most  of  his  way  in  the  steps  of  one,  at 
least,  who  has  gone  before  him. 

The  "  Advanced  Lessons  "  are  designed  to  continue  and  extend 
the  course  of  discipline  in  numbers,  which  is  begun  in  the  elemen- 
tary book  above  named.  Consequently  it  requires,  for  its  success- 
ful study,  an  acquaintance  with  the  elements,  as  taught  in  that 
work,  or  in  some  other  occupying  essentially  the  same  ground. 

In  all  the  mental  calculations  in  large  sums,  it  will  be  found  a 
uniform  characteristic  of  this  work  to  begin  with  the  highest  order 
of  numbers  in  the  sum,  — ■  hundreds  before  tens,  tens  before  units. 
In  this  way,  the  numbers  are  presented  in  the  same  order  in  which 
they  are  presented  in  the  common  usage  of  our  language.  In 
most  of  the  operations  of  written  Arithmetic,  however,  the  smallest 
number  is  taken  first ;  and  thus  a  method  is  pursued,  the  reverse 
of  what  the  genius  of  our  language  would  naturally  suggest. 
Another  advantage  of  taking  the  highest  numbers  first,  in  Mental 
Arithmetic  is,  that  we  thus  obtain  a  large  approximation  to  the 
final  answer,  at  the  first  step.  When  the  first  step,  however, 
as  in  written  addition,  or  multiplication,  furnishes  only  the 
units  of  the  answer,  leavingj  the  hundreds  or  thousands  still 
unknown,  only  a  minute  fraction  of  the  answer  is  at  first  ob- 
tained. It  is  too  plain  to  require  proof,^that  that  method  will  be 
most  interesting  and  gratifying  to  the  mind,  which  secures  the 
largest  portion  of  the  answer  at  the  first  step.    Another  advantage 


PREFACE.  V 

of  the  method  here  used,  is  found  in  the  fact,  that  we  naturally 
make  the  higher  order  the  standard,  and  the  lower  order  takes  its 
value  in  the  mind  from  a  comparison  with  the^higher,  as  a  certain 
part  of  it.  Thus  150  is  apprehended  by  the  mind,  as  one  hundred 
and  half  a  hundred.  This  is  not,  indeed,  the  method  of  acquir- 
ing the  idea  of  large  numbers,  but  the  method  of  combining  them 
after  the  idea  has  been  acquired ;  consequently,  it  is  the  legitimate 
method  of  instruction,  just  as  soon  as  the  pupil  is  qualified  to  enter 
on  the  study  of  such  combinations.  If,  now,  we  obtain  the  num- 
ber of  the  highest  order  first,  we  have  a  standard,  under  which  all 
the  succeeding  orders  naturally  fall,  and  from  a  comparison  with 
which  they  successively  take  their  value.  If  we  begin  with  units, 
however,  and  work  upward  through  the  higher  orders,  we  obtain 
no  standard ;  we  must  hold  the  successive  numbers  in  suspense, 
until  the  last  term  shall  furnish  the  nucleus  for  the  group, — the 
standard  under  which  all  the  lower  orders  shall  take  their  rank. 

It  is  on  the  basis  of  these  facts,  which  are  only  indications  of  the 
laws  of  the  mind,  that,  throughout  the  Mental  part  of  this  Arith- 
metic, the  author  has  in  all  operations,  taken  the  highest  order  of 
numbers  first.  The  increased  interest  which  the  persevering  use 
\  of  this  method  will  awaken  in  the  minds  of  pupils,  will  be,  to 
teachers,  a  better  commendation  of  it»-  correctness,  than  any  more 
•-extended  mental  analysis.  *-- 

There  are  other  features  of  the  Advanced  Lessons  which  are, 
perhaps,  sufficiently  distinctive  to  justify  their  mention  here  ;  but 
as  the  truest  test  of  a  school  book  is  its  use  in  the  school  room, 
the  work  is  referred  to  that  ordeal. 

The  Second  Part  contains  examples  in  "V^tten  Arithmetic  on 

all  the  most  important  rules.     They  are  designed  to  be  sufficiently 

numerous  to  lead  the  student  to  ready  and  accurate  practice  in 

ciphering.     In   this  Part  the   author  has  aimed  to  interest,  the 

1* 


Bcholar  by  furnishing  liim  with  natural  and  reasonable  questions, 
and  to  aid  both  teacher  and  scholar  by  arranging  them  progres- 
sively. 

The  rules  and  explanations  will,  probably,  be  found  sufficient, 

after  a  thorough  mastery  of  the  First  Part.  It  is  not  necessary 
that  the  pupil  complete  the  First  Part  before  beginning  the  Second. 
He  may  carry  on  both  Parts  at  the  same  time ;  but,  under  each 
particular  head,  the  mental  part  should  be  thoroughly  mastered 
before  the  written  examples  are  begun. 

The  answers  to  the  questions  in  the  Second  Part  are  given  in 
a  separate  work.  This  course  has  seemed  to  the  author,  on  the 
whole,  the  best,  notwithstanding  some  incidental  disadvantages 
that  may  arise  from  it.  It  will  enable  the  teacher  to  oversee  a 
much  larger  amount  of  work  in  Arithmetic,  than  he  could  other- 
wise attend  to. 

The  Key  will  be  bound  up  with  the  Arithmetic,  for  the  use  of 
teachers ;  and  such  copies  will  be  lettered  Teacher's  Copy. 

The  present  contains  a  considerable  number  of  examples  more 
than  the  Third  Edition,  but  no  change  In  the  numbering  of  the 
sections  or  of  the  examples,  to  occasion  inconvenience  to  the  teacher. 

To  aid  in  awakening  a  higher  interest  and  zeal  in  this  branch 
of  study,  the  author  will  offer  a  few  suggestions. 

Let  the  key  be  used  as  little  as  the  teacher's  necessities  will 
permit. 

Let  original  questions  be  proposed  by  the  teacher  in  connexion 
with  every  Section. 

Each  member  of  the  class  should  be  encouraged  to  propose 
original  questions  to  be  solved  by  the  class. 

It  will  often  be  useful,  especially  in  a  review,  to  alter  some  one 
figure  in  the  conditions  of  each  question.  This  often  produces  a 
happy  excitement,  and  gives  quite  a  new  zest  to  the  study. 

DuMMER  Academy,  April  18,  1846. 


CONTENTS. 


PAUT   FIRST. 

Section.  Page. 

Preface, 3 

^  Explanations, 11 

I.    Multiplication  of  Tens  and  UniA, 13 

n.    Multiplication  of  Tens  and  Units. —  Complement,  •  •  .16 

in.     Practical  Questions, 18 

IV*.    Division, 20 

V.     Time.  —  Linear  Measure, 25 

VI.     Federal  Money.  —  Sterling  Money.  —  Dry  Measure. 

—  Avoirdupois  Weight. —  Troy  Weight. —  Apothe- 
caries' Weight.  —  Cloth  Measure.-^Wine  Measure. 

—  Beer  Measure.  —  Measure  of  the  Circle, 33 

Vn.     Prime  numbers, 42 

VILl.    Multiplication  and  Division  of  Fractions. — .To  Find 

the  Divisors  of  Numbers, 48 

IX.  Multiplication  of  Fractions  by  Fractions.  —  Division 
of  Fractions  by  Fractions.  —  Addition  of  Frac- 
tions. —  To  find  a  Common  Denominator, 53 

X.     The  least  Common  Multiple, • 60 

XI.     Practical  Questions, •  •  62 

Xn.  Decimal  Fractions.  —  Addition  and  Subtraction  of 
Decimals.  —  Multiplication  of  Decimals. — Division 

of  Decimals, • 65 

Xm.    Keduction  of  Vulgar  Fractions  to  Decimals, 71 

XIV.    Interest.  —  Banking.  —  Discount.  —  Loss  and  Gain. 

—  Per  Centage, 75 


vm    ,  CONTENTS. 

XV.     Square  Measure, 81 

XVI.     Construction  of  the  Square.  —  Practical  Questions,-  •  -85 

XVn.    Practical  Questions  in  Square  Measure, 91 

XVIII.     Analysis  of  Problems, 95 

XIX.     Solid  Measure. —  Construction  of  the  Cube, 98 

XX.    Ratio.  —  Proportion.  —  Comparison  of  Similar  Sur- 
faces.—  Comparison  of  Similar  Solids, 103 

Notes  to  Part  First, "• 114 


PART    SECOND. 

4 

Numeration  of  whole  Numbers. — Numeration 

of  Decimals, * 117 

I.  Addition, 1 20 

II.  Subtraction, 122 

in.  Multiplication,   124 

IV.  Division, / 126 

V.  Reduction, 129 

VI.  Reduction, • 131 

VII.  Compound  Addition, 132 

Vin.  Compound  Subtraction, 134 

IX  Compound  Multiplication, ' 135 

X  Compound  Division, 136 

XI.  Miscellaneous  Examples, 137 

Xn.  Divisibility  of  Numbers, 138 

Xni.  Reduction  of  Fractions, 139 

XIV.  Change  of  Numbers  and  Fractions  to  Higher 

Terms, 141 

XV.  Multiplication  and  Division  of  Fractions, 142 

XVI.  Multiplication  and  Division  of  Fractions, 143 

XVn.  Addition  and  Subtraction  of  Fractions, 144 

XVIII.  Reduction  of  Denominate  Fractions, 145 

XIX.  Change  of  Denominate  Integers  to  Fractions,  •  •  •  146 


XX. 
XXL 


xxn. 
xxni. 

XXIV. 

XXV. 
XXVI. 

xxvn. 


xxvm. 

XXIX. 
XXX. 
XXXI. 

xxxn. 
xxxm. 

XXXIV. 

XXXV. 

XXXVI. 

xxxvn. 
xxxvin. 

XXXIX. 

XL. 

XLI. 

XLH. 

XLm. 

XLIV. 


CONTENTS.  IX 

Practical  Examples, 147 

Decimal  Fractions.  —  Addition  and  Subtraction. 

—  Multiplication  of  Decimals.  —  Division  of 
Decimals, 14.8 

Reduction  of  Vulgar  Fractions  to  Decimals.  — 

Repeating  and  Circulating  Decimals,  •  •  • 1^9 

Reduction  of  Denominate  Integers  to  Decimals,  -151 
To  find  the  Integral  Value  of  Denominate  Deci- 
mals,  151 

Practical  Examples, 152 

Practical  Questions  in  Vulgar  and  "Decimal  Frac- 
tions,   • 154 

Reduction   of  Currencies.  —  English   Currency. 

—  Federal  Money  to  Sterling.  —  Canada  Cur- 
rency. —  New  England  Currency.  —  New 
York  Currency.  —  Pennsylvania  Currency.*  •  -155 

Interest, 157 

Partial  Payments.  —  Annual  Interest, •  •  160 

Discount, • 163 

Banking, 1 64 

Loss  and  Gain.  —  Per  Centage, 165 

Alligation, 168 

Equation  of  Payments, 171 

Square  Measure, 172 

Duodecimals, 1 74 

Extraction  of  the  Square  Root, 1 75 

Extraction  of  the  Cube  Root, t^^.'^:i^'l79 

Proportion. —  Practical  Question^. —  Partnership,  181 

Arithmetical  Progression, 188 

Geometrical  Progression,  • 191 

Mensuration  of  Surfaces, 192 

Mensuration  of  Solids, 193 

Lllscellaneous  Theorems  and  Questions.  —  Spe-        ^.^^ 
clfic    Gravlt}^;^  MecTianlcnl    Po wcrsb-  —  Tho 
Lever.  —  The  Wheel  and  Axle.  —  The  Screw. 
—  Strength  of  Beams  to  resist  Fracture.  — 
Stiffness  of  Beams  to  resist  Flexure, 195 


X  CONTENTS. 

XLV.    Business  Forms  and  Instruments. — Promissory  Notes 

—  On  Demand,  witli  Interest ;  on  Time,  with  In- 
terest ;  on  Time,  without  Interest ;  Payable  by  In- 
stalments, with  Periodical  Interest.  —  Remarks  on 
Promissory  Notes.  —  Receipts  —  A  general  Form ; 
for  Money  paid  by  another  *  Person  ;  for  Money 
received  for  Another;  in  Part  of  a  Bond;  for 
Interest  due  on  a  Bond ;  on  Account ;  of  Papers. 

—  Order  at  Sight.  —  Order  on  Time.  —  Award 
by  Referees.  —  Letter  of  Credit.  —  Power  of  At- 
torney,   204 

XL VI.  On  the  Standard  of  Weights  and  Measures.  —  The 
English  System ;  Adopted  by  the  Government  of 
the  United  States.  —  French  Decimal  System.  — 
French  Long  Measure. — French  Square  Measure. 

—  French  Decimal  Weight, •  •  •  208 

XL VII.   Appendix, 213 


EXPLANATIONS 


1.  The  sign  =  indicates  equality;  as  7  times  3=21. 

2.  The  sign  -j-  indicates  addition;  as  15-j-7=22. 

3.  The  sign  —  placed  between  two  numbers,  indicates  that 
the  latter  number  is  to  be  taken  from  the  former ;  as  9 — 4 
t=5. 

The  larger  number  is  called  the  minuend ;  the  smaller,  the 
subtrahend. 

4.  The  sign  X  indicates  multiplication;  as  6X7=42. 
The  two  numbers  are  called  factors ;  the  number  multiplied 

is  called  the  multiplicand ;  the  number  by  which  it  is  multi- 
plied, the  multiplier. 

5.  The  sign  H-  indicates  that  the  number  placed  before  it, 
is  to  be  divided  by  the  number  after  it ;  as  15-r-5=3. 

The  number  to  be  divided  is  called  the  dividend ;  the  num- 
ber by  which  it  is  divided  is  called  the  divisor. 

6.  When  a  number  is  multiplied  by  itself,  the  product  is 
called  the  second  power  of  that  number,  or  the  square  of  it ; 
as  2  X  2=4,  which  is  the  second  power,  or  the  square  of  2 ; 
so  9  is  the  square  of  3  ;  25  the  square  of  6. 

7.  When  a  number  is  multiplied  hy  itself,  so  as  to  be  taken 
3  times  as  a  factor,  the  product  is  called  the  3d  power,  or  the 
cube  of  the  number ;  thus  8  is  the  cube  of  2,  for  it  is  formed 
by  multiplying  2X2X2;  27,  or  3X3X3,  is  the  cube  or  third 
power  of  3 ;  125,  or  5x5X5,  is  the  third  power  of  5.  The 
number  thus  used  as  a  factor,  is  called  the  root  of  the  power ; 
thus  3  is  the  square  root  of  9,  and  the  cube  root  of  27 ;  5  is 
the  square  root  of^S. 

The  number  of  the  power  may  be  expressed  by  a  small 
figure,  thus  2^  is  the  3d  power  of  2;  3  2  is  the  2d  power  of  3 ; 
53  is  the  3d  power  of  5. 


12  EXPLANATIONS. 

Ah  angle 'is  formed  when  two  lines  meet,  run- 
ning in  different  directions. 

A  triangle  is  a  figure  bounded  by  three  straight 
lines.  It  is  called  a  triangle,.because  it  has  three 
angles.  An  equilateral  triangle  has  all  its  sides 
equal. 

A  right  angle  is  formed  when  one  line  meets 
_,  another,  making  the  angles  on  both  sides  equal. 


0A  square  is  a  four-sided  figure,  the  sides  of  which 
are  all  equal,  and  the  angles  of  which  are  right  angles. 
The  diagonal  divides  it  into  two  equal  parts. 

A  rectangle  is  a  four-sided  figure,  the  oppo- 
site sides  of  which  are  equal,  and  the  angles  of 
which  are  right  angles.  The  diagonal  divides 
it  into  two  equal  parts. 

A  parallelogram  is  a  four-sided  figure  the 
opposite  sides  of  which  are  equal  and  parallel. 
The  diagonal  divides  it  into  two  equal  parts. 

A  circle  is  a  figure  bounded  by  a  curved  line, 
called  the  circumference,  every  part  of  which  is 
equally  distant  from  the  centre. 

A  straight  line  from  the  centre  to  the  circumference  is 
called  the  radius. 

The  diameter  is  a  line  drawn  from  side  to  side  of  the  cir- 
cle, through  the  centre.  It  follows  that  the  diameter  is  equal 
to  twice  the  radius. 

Any  portion  of  the  circumference  considered  by  itself  is 
called  an  arc. 

A  sector  of  a  circle  is  a  portion  of  it  bounded  by  two  ra- 
dii and  the  arc  between  them. 

A  sphere  is  a  solid  bounded  by  a  curved  surface  every  part 
of  which  is  equally  distant  from  the  centre  of  the  solid. 


O 


MENTAL    ARITHMETIC 


PART    FIRST 


SECTION    I. 


MULTIPLICATION  OF  TENS  AND  UNITS. 

1.  A  man  drove  six  oxen  to  market,  and  sold  three  of  them 
for  50  dollars  apiece.     What  did  they  come  to  ? 

Three  times  50  are  150.     Ans.  150  dollars. 

He  sold  the  remaining  three  for  52  dollars  apiece.  What 
did  they  come  to  ? 

Three  times  50  are  150,  and  three  times  2  are  6,  which 
added  to  150  makes  156.     Ans.  156  dollars. 

What  did  they  all  come  to  ? 

Twice  100  is  200,  and  twice  50  is  100,  which  added  to  200 
makes  300,  and  6  added  to  300  makes  306.    Ans.  306  dollars. 

2.  A  merchant  bought  45  barrels  of  flour  for  6  dollars  a 
barrel.     What  did  it  come  to  ? 

6  times  40  are  240,  and  6  times  5  are  30 ;  30  added  to  240 
makes  270.     Ans.  270  dollars. 

He  bought  75  barrels  more  at  5  dollars  a  barrel.  What 
did  it  come  to  ? 

5  times  70  are  350 ;  5  times  5  are  25,  which  added  to  350 
makes  375.     Ans.  375  dollars. 

What  did  all  the  flour  come  to  ? 

300  and  200  are  500,  70  and  70  are  140,  which  added  to 
500  makes  640,  and  5  are  645.     Ans.  645  dollars. 

3.  What  will  87  barrels  of  flour  come  to  at  6  dollars  a 
barrel  ? 

6  times  80  are  480,  and  6  times  7  are  42,  which  added  to 
480  makes  522.    Ans.  522  dollars. 

2 


14 


MENTAL    ARITHMETIC. 


4.  What  are  7  times  68  ?     What  are  8  times  72  ? 
What  are  9  times  84  ?     What  are  4  times  96  ? 
8  tunes  64?  5  times  72  ?  7  times  83  ?  5  times  79  ? 
4  times  98  ?  3  times  81  ?  6  times  73  ?  6  times  86  ? 

The  preceding  examples  will  show  the  importance  of  being 
able  readily  to  multiply  tens  by  units.  This  becomes  easy, 
after  acquiring  the  Multiplication  Table.  It  may  be  con- 
nected with  a  review  of  the  Multiplication  Table  in  the  follow- 
ing manner. 

Twice  1  are  how  many  ?     Twice  10  are  how  many  ? 

Twice  2  are  how  many  ?     Twice  20  are  how  many  ? 

Twice  3?    Twice  30?     Twice    4?    Twice    40? 

Twice  5?     Twice  50?     Twice    6?     Twice    60? 

Twice  7  ?     Twice  70  ?     Twice    8  ?     Twice    80  ? 

Twice  9  ?    Twice  90  ?     Twice  10  ?    Twice  100  ? 


3  times  1  ? 
3  times  3  ? 
3  times  5  ? 
3  times  7  ? 

3  times  9  ? 

4  times  1  ? 
4  times  3  ? 
4  times  5  ? 
4  times  7  ? 

4  times  9  ? 

5  times  1  ? 
5  times  3  ? 
5  times  5  ? 
5  times  7  ? 

5  times  9  ? 

6  times  1  ? 
6  times  3  ? 
6  times  5  ? 
6  times  7  ? 

6  times  9  ? 

7  times  1  ? 
7  times  3  ? 
7  times  5  ? 
7  times  7  ? 
7  times  9  ? 


3  times  10  ? 
3  times  30  ? 
3  times  50  ? 
3  times  70  ? 

3  times  90  ? 

4  times  10  ? 
4  times  30  ? 
4  times  50  ? 
4  times  70  ? 

4  times  90  ? 

5  times  10  ? 
5  times  30  ? 
5  times  50  ? 
5  times  70  ? 

5  times  90  ? 

6  times  10  ? 
6  times  30  ? 
6  times  50  ? 
6  times  70  ? 

6  times  90  ? 

7  times  10  ? 
7  times  30  ? 
7  times  50  ? 
7  times  70  ? 
7  times  90  ? 


3  times  2  ? 
3  times  4  ? 
3  times  6  ? 
3  times    8  ? 

3  times  10  ? 

4  times  2  ? 
4  times  4  ? 
4  times  6  ? 
4  times  8  ? 

4  times  10  ? 

5  times  2  ? 
5  times  4  ? 
5  times  6  ? 
5  times  8  ? 

5  times  10  ? 

6  times  2? 
6  times  4  ? 
6  times  6  ? 
6  times  8  ? 

6  times  10  ? 

7  times  2  ? 
7  times  4  ? 
7  times  6  ? 
7  times  8  ? 
7  times  10  ? 


3  tunes  20  ? 
3  times  40  ? 
3  times  60  ? 
3  times    80  ? 

3  times  100  ? 

4  times  20? 
4  times  40  ? 
4  times  60  ? 
4  times  80? 

4  times  100  ? 

5  times  20  ? 
5  times  40  ? 
5  times  60  ? 
5  times  80  ? 

5  times  100  ? 

6  times  20? 
6  times  40  ? 
6  times  60? 
6  tunes  80? 

6  times  100  ? 

7  times  20  ? 
7  times  40  ? 
7  times  60  ? 
7  times  80? 
7  times  100  ? 


MULTIPLICATION    OF  TESS   AND    UNITS. 


15 


8  times  1  ?  8  times  10  ?  8  times    2  ?  8  times    20  ? 

8  times  3  ?  8  times  30  ?  8  times    4  ?  8  times    40  ? 

8  times  5  ?  8  times  50  ?  8  times    6  ?  8  times    60  ? 

8  times  7  ?  8  times -70  ?  8  times    8  ?  8  times    80  ? 

8  times  9  ?  8  times  90  ?  8  times  10  ?  8  times  100  ? 


9  times 
9  times 
9  times 
9  times 
9  times 

10  times 
10  times 
10  times 
10  times 

10  times 

11  times 
11  times 
11  times 
11  times 
11  times 


1? 

3? 
5? 

7? 
9? 

1? 
3? 
5? 

7? 
9? 

1? 

3? 
5? 

7? 
9? 


9  times 
9  times 
9  times 
9  times 
9  times 

10  times 
10  times 
10  times 
]  0  times 

10  times 

11  times 
11  times 
11  times 
11  times 
11  times 


10? 
30? 
50? 
70? 
90? 

10? 
30? 
50? 
70? 
90? 

10? 
30? 
50? 

70? 
90? 


11  times  11?    11  times  110? 


12  times 
12  times 
12  times 
12  times 
12  times 


1? 
3? 
5? 

7? 
9? 


12  times 
12  times 
12  times 
12  times 
12  times 


10? 
30? 
50? 
70? 
90? 


12  times  11  ?    12  times  110  ? 


9  times  2  ? 
9  times  4  ? 
9  times  6  ? 
9  times  8  ? 
9  times  10  ? 

10  times  2  ? 
10  times  4  ? 
10  times  6  ? 
10  times    8  ? 

10  times  10  ? 

11  times  2  ? 
11  times  4  ? 
11  times  6  ? 
11  times  8  ? 
11  times  10  ? 

11  times  12? 

12  times  2  ? 
12  times  4  ? 
12  times  6  ? 
12  times  8? 
12  times  10  ? 
12  times  12  ? 


9  times  20  ? 
9  times  40  ? 
9  times  60  ? 
9  times  80  ? 
9  times  100  ? 

10  times  20  ? 
10  times  40  ? 
10  times  60  ? 
10  times    80? 

10  times  100  ? 

11  times  20  ? 
11  times  40? 
11  times  60  ? 
11  times  80? 

11  times  100? 

12  times  120  ? 

12  times  20  ? 
12  times  40  ? 
12  times  60  ? 
12  times  80? 
12  times  100? 
12  times  120  ? 


A  number  which  contains  another  number  a  certain  number 
of  times,  is  a  multiple  of  that  number. 

Thus  6  is  a  multiple  of  2 ;  15  of  3  ;  28  of  7.* 

Name  all  the  multiples  of  2,  from  2  to  60. 

Name  the  multiples  of  20,  from  20  to  600. 

What  are  the  multiples  of  3  up  to  75  ?  of  30  up  to  750  ? 

What  are  the  multiples  of  4  up  to  80  ?  of  40  up  to  800  ? 

What  are  the  multiples  of  5  up  to  100  ?  of  50  up  to  1000  ? 
of  6  to  72  ?  of  60  to  720  ?  of  7  to  84  ?  of  70  to  840  ?  of  8 
to  96  ?  of  80  to  960  ?  of  9  to  108  ?  of  90  to  1080  ?  of  10  to 
120  ?  of  100  to  1200  ? 


*  See  Note  1,  at  the  end  of  Part  First- 


16 


MENTAL    ARITHMETIC. 


SECTION    II. 


MULTIPLICATION  OF  TENS  AND  UNITS.  — COMPLEMENT. 

1.  What  will  17  tons  of  liay  come  to  at  8  dollars  a  ton  ? 
Aus.  8  times  10  are  80,  and  8  times  7  are  56 ;  56  added 

to  80  makes  136.     136  dollars. 

2.  What  will  37  pounds  of  sugar  come  to  at  9  cents  a  pound  ? 

3.  A  man  drove  87  sheep  to  market,  and  sold  them  for  6 
dollars  apiece.     What  did  they  come  to  ? 

4.  A  man  travelled  on  foot  8  days  ;  he  travelled  29  miles 
each  day.     How  many  miles  did  he  travel  in  all  ? 

In  each  of  the  above  examples  the  second  j^roduct  when 
added  to  the  first  makes  a  sum  e^^eeding  the  next  even  hun- 
dred: thus,  in  the  1st  ex.  — 80+56 ;  in  the  2d,  270+63  ;  in 
the  3d,  480+42 ;  in  the  4th,  160+72. 

In  order  to  perform  such  examples  with  ease,  quickness, 
and  without  mistake,  each  step  in  the  process  should  be  made 
the  subject  of  distinct  practice.  To  illustrate  these  steps  by 
the  first  example,  80+56,  the  first  thing  to  be  done  is  *to 
think  of  the  number  which  must  be  added  to  80  to  make  100, 
namely,  20  ;  the  next  is  to  take  this  20  from  the  56,  aad  what 
remains,  —  36,  —  will  belong  to  the  next  hundred. 

The  number  which  in  such  cases  must  be  added  to  a  given 
number  to  make  up  an  even  hundred  may  be  called  the  Com- 
plement of  that  number.  Thus  the  complement  of  80  is  20 ; 
of  60,  40  ;  of  90,  10  ;  of  56,  44.  What  is  the  complement  of 
10  ?  30  ?  50  ?  70  ? 

*  What  is  the  complement  of 


10? 

20? 

30? 

40? 

50? 

60? 

70? 

80  ?~     90? 

11? 

21? 

31? 

41? 

51? 

61? 

71? 

81?      91? 

12? 

22? 

32? 

42? 

52? 

62? 

72? 

82?      92? 

13? 

23? 

33? 

43? 

53? 

63? 

73? 

83?      93? 

14? 

24? 

34? 

44? 

54? 

64? 

74? 

84?      94? 

15? 

25? 

35? 

45? 

55? 

65? 

75? 

85?      95? 

16? 

26? 

36? 

46? 

56? 

66? 

76? 

86?      96? 

17? 

27? 

37? 

47? 

57? 

67? 

77? 

87?      97? 

18? 

28? 

38? 

48? 

58? 

68? 

78? 

88?      98? 

19? 

29? 

39? 

49? 

59? 

69? 

79? 

89?      99? 

How  many   are 
25+83?   36+71? 
37+84?   45+76? 

40+76?    80+34? 
45+82?    56+73? 
88+37?    94+17? 

70+91 
43+82 
76+87 

?    90+17? 
t?    95+36? 

'  ? 

*  See  Note  2. 

MULTIPLICATION    OF    TENS   AND    UNITS. 


17 


*  How  many  are 

How  many  are 

12X2,3,4,5,6,7,8,9,10? 

46X2,3,4,5,6,7,8,9,10? 

13X2,3,4,5,6,7,8,9,10? 

47X2,3,4,5,6,7,8,9,10? 

14X2,3,  4,5,  6,  7,  8/9,  10? 

4^X2,3,4,5,6,7,8,9,10? 

15X2,3,4,5,6,7,8,9,10? 

49X2,3,4,5,6,7,8,9,10? 

16X2,3,4,5,6,7,8,9,10? 

50X2,3,4,5,6,7,8,9,10? 

17X2,3,4,5,6,7,8,9,10? 

51X2,3,4,5,6,7,8,9,10? 

18X2,3,4,5,6,7,8,9,10? 

52X2,3,4,5,6,7,8,9,10? 

19X2,3,4,5,  6,7,8,9,10? 

53X2,3,4,5,6,7,8,9,10? 

20X2,3,4,5,6,7,8,9,10? 

54X2,3,4,5,6,7,8,9,10? 

21X2,3,4,5,6,7,8,9,10? 

55X2,3,4,5,6,7,8,9,10? 

22X2,3,4,5,6,7,8,9,10? 

56X2,3,4,5,6,7,8,9,10? 

23X2,3,4,5,  6,7,8,9,10? 

57X2,3,4,5,6,7,8,9,10? 

24X2,3,4,5,6,7,8,9,10? 

58X2,3,4,5,6,7,8,9,10? 

25X2,3,4,5,6,7,8,9,10? 

59X2,3,4,5,6,7,8,9,10? 

26X2,^3,4,5,6,7,8,9,10? 

60X2,3,4,5,6,7,8,9,10? 

27X2,3,  4,  5,  6,7,8,9,  10? 

61X2,3,4,5,6,7,8,9,10? 

28X2,3,4,5,  6,7,8,9,10? 

62X2,3,4,5,6,7,8,9,10? 

29X2,3,4,5,6,7,8,9,10? 

63X2,3,4,5,  6,7,8,9,  10? 

30X2,3,4,5,6,7,8,9,10? 

64X2,3,4,5,6,7,8,9,10? 

31X2,3,4,5,6,7,8,9,10? 

65X2,3,4,5,  6,7,8,9,10? 

32X2,3,4,5,6,7,8,9,10? 

66X2,3,4,5,6,7,8,9,10? 

33X2,^,4,5,  6,7,8,9,10? 

67X2,  3,  4,5,  6,7,8,  9,  10? 

34X2,3,4,5,6,7,8,9,10? 

68X2,3,4,5,6,7,8,9,10? 

35X2,3,4,5,6,7,8,9,10? 

69X2,3,4,5,  6,7,8,9,10? 

36X2,3,4,5,6,7,8,9,10? 

70X2,3,4,5,6,7,8,9,10? 

37X2,3,4,5,6,7,8,9,10? 

71X2,3,4,5,6,7,8,9,10? 

38X2,3,4,5,  6,7,8,9,  10? 

72X2,3,4,5,6,7,8,9,10? 

39X2,3,4,5,6,7,8,9,10? 

73X2,3,4,5,6,7,8,9,10? 

40X2,3,4,5,5,7,8,9,10? 

74X2,3,4,5,6,7,8,9,10? 

41X2,  3,it,  5,  6,  7,- 8,  9,  10? 

75X2,3,  4,5,  6,  7,8,  9,  10? 

42X2,3,4,5,6,7,8,9,10? 

76X2,3,4,5,6,7,8,9,10? 

43X2,3,4,5,  6,7,8,9,10? 

77X2,3,4,5,6,7,8,9,10? 

44X2,3,4,5,  6,7,8,9,  10? 

78X2,3,4,5,6,7,8,9,10? 

45X2,3,4,5,6,7,8,9,10? 

79X2,3,4,5,6,7,8,9,10? 

To  multiply  any  number  less  than  10  by  11,  repeat  the  fig- 
ure expressing  the  number :  as  3  times  11  is  33,  4X11=44. 

To  multiply  by  11  any  number  of  two  figures.  Think  of 
the  first  figure,  then  of  the  sum  of  the  two  figures,  then  of  the 
last  figure.     These   three  figures  will  express  the  answer. 


2* 


^  Note  3. 


18  MENTAL    ARITHMETIC. 

Thus  11 X  23  ;  the  first,  2  ;  the  sum  of  the  two,  5  ;  the  last,  3. 
Ans.  253.     11X24=264,  11X32=352,  11X43=473. 

Remember,  if  the  sum  of  the  two  is  as  much  as  10,  you 
must  increase  the  first  figure  by  one. 

How  many  are  11X26?  11X28?  11X29?  11X41? 
11X43?  11X45?.  11X61?  11X62?  11X64?  11X71? 
11X73?  11X81?  11X94?  11X75?  11X86?  11X89? 
11X82?    11X84? 


SECTIO^N    III. 

-    PRACTICAL   QUESTIONS. 

1.  If  a  rail-road  car  travels  23  miles  in  one  hour,  how  far 
will  it  travel  in  9  hours  ? 

2.  If  a  horse  travels  38  miles  in  one  day,  how  far  will  he 
travel  in  6  days  ? 

3.  If  a  man  earns  14  dollars  a  month,  how  much  will  he 
earn  in  7  months  ? 

4.  If  a  man  spends  6  cents  a  day  for  ardent  spirit,  how 
much  will  that  amount  to  in  10  days  ?  How  much  in  30 
days  ?  How  much  in  300  days  ?  How  much  in  60  days  ? 
How  much  in  5  days  ?     How  much  in  365  days  ? 

5.  If  a  man  earns  10  cents  in  an  hour,  and  works  12  hours 
in  a  day,  how  much  will  he  earn  in  a  week,  there  being  6 
working  days  in  a  week  ?  How  much  in  10  weeks  ?  How 
much  in  50  weeks  ? 

6.  If  a  scholar  in  school  is  idle  18  minutes  in  the  forenoon, 
and  18  minutes  in  the  afternoon,  how  much  time  will  he  lose 
in  a  week,  if  there  are  6  forenoons,  and  4  afternoons  of  school 
time  in  a  week  ? 

7.  If  a  town  is  6  miles  long,  and  5  miles  broad,  how  many 
square  miles  does  it  contain  ?  If  there  are  40  inhabitants  on 
every  square  mile,  how  many  inhabitants  does  the  town  con- 
tain ?  40  times  30.  4  times  30  are  120.  40  times  30  are  10 
times  as  many.  If  one  in  12  of  the  inhabitants  were  able- 
bodied  men,  how  many  able-bodied  men  would  there  be  ?  If 
one  in  6  are  able-bodied  men,  how  many  such  are  there  ? 

8.  What  will  146  yards  of  broadcloth  come  to  at  5  dollars 
a  yard  ? 


PRACTICAL    QUESTIONS.  ^  19 

9.  What  will  86  yards  of  broadcloth  come  to  at  6  dollars 
and  a  half  a  yard  ? 

10.  What  will  740  barrels  of  flour  come  to  at  6  dollars 
a  barrel  ?  at  5  dollars  a  barrel  ?  at  5  dollars  and  a  half  a 
barrel ? 

11.  What  will  33  gallons  of  molasses  come  to  at  31  cents  a 
gallon  ?    at  34  cents  a  gallon  ?    at  40  cents  a  gallon  ? 

12.  What  "s\dll  38  pounds  of  coffee  come  to  at  14  cents  a 
pound  ?    at  16  cents  a  pound  ? 

13.  If  a  room  is  14  feet  long  and  9  feet  high,  how  many 
square  feet  are  there  in  one  of  the  side  walls  ?  How  many 
in  both  the  side  walls  ?  If  the  same  room  is  13  feet  wide, 
how  many  square  feet  are  there  in  one  of  its  end  walls  ? 
How  many  in  both  its  end  walls.  How  many  square  feet  in 
the  ceiling  ?  • 

14.  A  man  wishes  to  know  how  many  shingles  he  must  buy 
in  order  to  shingle  his  house.  His  touse  is  40  feet  long,  and 
it  is  18  feet  from  the  eaves  to  the  ridgepole.  How  many 
square  feet  are  there  in  one  half  of  the  roof?  How  many  in 
the  whole  roof? 

One  thousand  shingles  will  cover  10  feet  square,  how  many 
thousand  shingles  will  cover  the  roof?  If  shingles  cost  4  dol- 
lars a  thousand,  how  much  must  be  paid  for  shingles  enough 
to  cover  the  roof?  If  the  labor,  the  boards,  and  the  nails, 
added  together,  cost  as  much  as  the  shingles,  what  will  be  the 
whole  expense  of  boarding  and  shingling  the  roof? 

15.  What  are  8  J  tons  of  hay  worth,  at  13  dollars  a  ton  ? 

16.  If  one  acre  of  ground  produce  65  bushels  of  corn,  how 
much  would  grow  on  9  acres  ? 

17.  If  an  acre  of  ground  produce  228  bushels  of  potatoes, 
how  many  bushels  would  grow  on  5  acres  ? 

18.  If  standing  wood  is  worth  2  dollars  a  cord,  what  is  the 
value  of  the  wood  on  7  acres,  each  of  which  furnishes  18  cords  ? 

19.  If  there  are  200  families  in  a  town,  and  each  family 
consumes  12  cords  of  wood  annually,  how  many  cords  are 
used  in  the  town  each  year  ? 

What  is  the  whole  value  of  the  wood  at  3  J^  dollars  a  cord  ? 
How  much  money  will  be  saved  in  the  town  if  each  family 
bums  2  cords  less  than  before  ? 


20  MENTAL    ARITHMETIC- 


SECTION    IV. 

DIVISION. 

1.  What  is  one  half  of  20?  of  40?  of  60  ?  of  80?  of  100? 
of  120?  of  140?  of  160? 

2.  What  is  one  half  22?  of  42?  of  62  ?  of  82?  of  102? 
of  112?  of  122?  of  142?  of  162?  of  182? 

3.  What  is  one  half  of  44?  64?  86?  48?  66?  28?  84? 
68?  46?  24?  26?  62? 

4.  What  is  one  half  of  70  ?  divide  it  into  60  and  10. 
What  is  one  half  of  90?  divide  it  into  80  and  10. 

What  is  one  half  of  50  ?  of  30  ?  of  1 10  ?  of  130  ?  of  150  ? 

5.  What  is  one  half  of  32  ?  of  54  ?  divide  it  into  50  and  4. 
.     What  is  one  half  of  76?  of  74?  of  78?  of  96?  of  98?  of 

92?  of  94?  of  72?  of  7S?  of  58?  of  56? 

6.  What  is  one  half -of  43?  One  half  of  40  is  20.  One 
half  of  3  is  1^,  this  added  to  20  makes  21^. 

What  is  one  half  of  47  ?  of  49  ?  of  63  ?  of  65  ?  of  67  ?  of 
69?  of83?  of85?  of87?  of89? 

7.  What  is  one  half  of  33  ?  divide  into  30  and  3. 

What  is  one  half  of  35?  37?  39?  51?  53?  55?  57?  59? 
of  71?  of  73?  of  75?  of  77?  of  79?  of  91?  of  93?  of  95? 
of  97?  of  99? 

8.  What  is  one  half  of  367  ?  divide  into  300,  60  and  7. 
What  is  one  half  of  674?  of  895?  of  724?  of  632?  of 

945?  of  424?  of  688?  of  546?  of  392? 

We  can  now  find  a  very  quick  way  of  multiplying  any 
number  by  5.  Take  one  half  the  number :  multiply  that  by 
10.  We  will  take  the  numbers  in  question  2,  and  multiply 
them  by  5  in  this  way. 

Multiply  22  by  5  :  half  of  22  is  11,  and  ten  times  11  is  110. 

Multiply  42  by  5  :  half  of  42  is  21 :  ten  times  that  is  210. 

Multiply  62  by  5  :  half  of  62  is  31  ;  310. 

Multiply  82  by  5  :  half  is  41 :  410. 

Multiply  102  by  5  :  half  is  51 :  510. 

Multiply  112  by  5  :  half  is  56  :  560. 

Multiply  122  by  5  :  half  is  61 :  010. 

Multiply  142  by  5  :  half  is  71 :  710. 

Multiply  162  by  5  :  half  is '81 :  810. 

Multiply  182  by  5  :  half  is  91 :  910. 


DIVISION.  21 

9.  Multiply  by  5  in  this  way  the  numbers  in  question  3. 
44.  64.  86.  48.  QQ.  28.  84.  68.  46.  24.  26.  62. 

Tou  can,  if  you  wish,  perform  these  examples  by  both 
methods,  and  thus  prove  the  work  correct. 

Multiply  862  by  5  :  half  is  431 :  4310.  '  Multiply  672  by 
5:  half  is  336:  3360. 

10.  Multiply  686  by  5.  748  by  5.  932  by  5.  896  by  5. 
1262  by  5. 

If  the  number  to  be  multiplied  is  an  odd  number,  so  that 
half  of  it  will  show  tbe  fraction  ^,  this,  when  you  multiply  by 
10,  will  become  5  :  for  ten  halves  are  5. 

Multiply  781  by  5  :  half  is  390 J:  3905  Ans. 

11.  Multiply  963  by  5  :  half  is  481^ :  4815.  Multiply  845 
by  5.     381  by  5.    953  by  5.   845  by  5.    637  by  5.   429  by  5. 

12.  What  is  one  fourth  of  40  ?  one  fourth  of  80  ?  one  fourth 
of  120  ?  One  fourth  of  12  is  3,  a  fourth  of  120  is  10  times  as 
much,  30.  What  is  one  fourth  of  160  ?  one  fourth  of  200? 
of  240  ?  of  280  ?  of  320  ?  of  360  ?  of  400  ? 

13.  What  is  one  fourth  of  60  ?  Take  half  of  it ;  then  half 
of  that  half;  half  of  60  is  30,  half  of  30  is  15.  What  is  one 
fourth  of  100?  one  fourth  of  140?  one  fourth  of  180?  one 
fourth  of  220  ?  one  fourth  of  260  ?  one  fourth  of  300  ? 

14.  Another  way  of  finding  one  fourth  of  the  numbers  in 
the  last  example,  is  as  follows : 

What  is  one  fourth  of  60  ?     Divide   60  into  40  and  20 ; 
one  fourth  of  40  is  10 ;  one  fourth  of  20  is  5.     15. 
What  is  one  fourth  of  100  ?  divide  into  80-f20. 
What  is  one  fourth  of  140?  divide  into  120+20. 
What  is  one  fourth  of  180?  divide  into  160-|-20,  &c. 

15.  What  is  one  fourth  of  30?  of  50?  of  70? 
Find  the  best  way  of  answering  these,  for  yourself. 
What  is  one  fourth  of  90?  of  110?  of  130?  of  150?  of 

170?  of  190?  of  210?  of  230?  of  250? 

16.  What  is  one  fourth  of  76  ?  divide  the  number  into  40 
and  36.  What  is  one  fourth  of  96?  divide  the  number  into 
80  and  16? 

What  is  one  fourth  of  52  ?  of  64  ?  of  84  ? 

-17.  What  is  one  fourth  of  368?  There  are  several  ways 
of  dividing  this  number.  First,  into  200+100+60+8;  a 
second  way  would  be,  into  200+1 60+8  ;  another  way,  into 
320+48.  This  is  shorter  than  either  of  the  former.  A  bet- 
ter division  still  is  into  360+8. 

What  is  one  fourth  of  496  ?     Into  what  different  sets  of 


22  MENTAL   ARITHMETIC. 

numbers,  each  divisible  by  4,  can  you  divide  this  ?  What  is 
one  fourth  of  964?  of  336?  of  836?  596?  472?  1324? 
1728?  2236? 

18.  What  is  one  tenth  of  10?  of  20?  of  30?  of  40?  of  50? 
of  60?  70?  80?  90?  100?  110?  120?  130?  140?  150? 

19.  What  is  one  tenth  of  5  ?  Ans.  5  tenths  of  one,  or  j% 
equal  to  J. 

What  then  is  one  tenth  of  15  ?  of  25  ?  35  ?  45  ?  55  ?  65  ? 
75?  85?  95?  14?  17?  36?  47?  52?  91?  43?  28?  65? 
86?  47? 

20.  What  is  one  fifth  of  25?  40?  45?  50?  55?  60?  65? 
70  ?  divide  70  into  50  and  20.  Of  75  ?  divide  into  50  and 
25.  Of  80  ?  divide  into  50  and  30.  Of  85  ?  of  90  ?  of  95  ? 
of  100? 

21.  What  is  one  fifth  of  64?  of  82?  of  91?  of  67?  of  73? 
of  59?  of  63?  of  72?  of  78?  of  83?  of  87? 

22.  What  is  one  fifth  of  140?  of  385?  of  260?  of  480? 
of  390  ?  of  580  ?  of  470  ?  of  865  ?  of  395  ? 

23.  The  following  is  a  short  way  of  dividing  a  number  by 
5:  Take  one  tenth  of  the  number  and  double  it.  That  of 
course  gives  2  tenths,  which  is  equal  to  one  fifth.  Take  the 
numbers  in  the  last  example,  and  divide  by  5  in  this  way. 
One  fifth  of  140 ;  one  tenth  is  14,  double  that  is  28.  One 
fifth  of  385  ;  one  tenth  is  38  and  5  tenths,  twice  that  is  77. 
One  fifth  of  260 ;  one  tenth  is  26,  twice  26  is  52.  One  fifth 
of  480 ;  one  tenth  is  48,  twice  that  is  96.  What  is  one  fifth 
of  390?  of  580?  470?  865?  395? 

24.  The  following  is  a  short  way  of  multiplying  a  number 
by  25.  Take  one  fourth  of  the  number ;  multiply  that  by 
100.  This  will  give  100  fourths,  which  are  equal  to  25  whole 
ones. 

Multiply  40  by  25 ;  one  fourth  is  10,  one  hundred  times 
that  are  1000.     Multiply  60  by  25 ;  one  fourth  is  15,  1500. 
Multiply  80  by  25  ;  one  fourth  is  20,  2000. 
Multiply  120  by  25 ;  a  fourth  is  30,  3000. 
Multiply  112  by  25 ;  a  fourth  is  28,  2800. 
Multiply  116  by  25  ;  a  fourth  is  29,  2900. 

25.  Multiply  22  by  25 ;  one  fourth  is  5  and  a  half;  100 
times  this  are  5  hundred  and  half  a  hundred,  550. 

Multiply  26  by  25 ;  one  fourth  is  6^,  650. 
Multiply  28  by  25  ;  one  fourth  is  7,  700, 
Multiply  30  by  25 ;  32  by  25  ;  34  by  25  j  36  by  25  ;  40 
by  25. 


DIVISION.  23 

Multiply  42  hj  25 ;  44  by  25  ;  46  by  25 ;  48  by  25 ;  50 
by  25. 

26.  Multiply  13  by  25 ;  one  fourth  is  3  and  one  fourth ; 
one  hundred  times  this  i3  300,  and  one  fourth  of  a  hundred,  or 
25,  325. 

Multiply  15  by  25;  one  fourth  is  3.  and  three  fourths,  one 
hundred  times  this  are  3  hundred  and  3  fourths  of  a  hundred, 
or  75,  375. 

Multiply  17  by  25  ;  19  by  25  ■;  21  by  25  ;  23  by  25 ;  27 
by  25  ;  29  by  25  ;  31  by  25  ;  33  by  25  ;  35  by  25. 

27.  Multiply  116  by  25  ;  one  fourth  is  29,  2900. 
Multiply  117  by  25 ;  one  fourth  is  29J,  2925. 
Multiply  121  by  25  ;  87  by  25 ;  156  by  25  ;  960  by  25. 
28.*  What  is  one  third  of  60  ?  of  90?   of  120?  of  15?  of 

150  ?  of  45  ?  of  450  ? 

What  is  one  third  of  18?  of  180?  of  21?  210?  of  36?  of 
360?of30?of390? 

What  is  one  third  of  72  ?   divide  into  60  and  12.  " 

Wliat  is  one  third  of  54  ?  of  85  ?  of  98  ? 

What  is  one  sixth  of  60  ?  of  80  ?  divide  into  60  and  20.  Of 
74?  of84?of  96?oflOO? 

What  is  one  sixth  of  12  ?  of  120  ?  of  130  ?  of  140  ?  of  144  ? 

What  is  one  sixth  of  18  ?  of  180  ?  of  200  ?  of  210  ?  of  220  ? 

What  is  one  sixth  of  384  f  of  492  ?  divide  into  480  and  12. 
Of  555  ?  divide  into  540  and  15.     Of  620  ?  of  726  ?  of  947  ? 

29.  What  are  the  two  factors  of  18  ?  of  180  ? 

What  are  the  two  factors  of  27  ?  of  270  ?  of  22  ?  of  220  ? 
of  35  ?  of  350  ?  of  54  ?  of  540  ?  of  45  ?  of  450  ?  of  21  ?  of  210  ? 
of  28  ?  of  280  ?  of  42  ?  of  420  ? 

30.  What  two  numbers  multiplied  together  will  produce 
24? 

What  other  two  factors  wiU  produce  24?  What  other  two  ? 

What  two  factors  will  produce  240  ?  What  other  two  ? 
What  others  ? 

What  two  factors  will  produce  30  ?     What  others  ? 

What  two  factors  will  produce  300  ?     What  others  ? 

What  two  factors  will  produce  18  ?     Wliat  others  ? 

What  two  will  produce  180  ?     What  others  ? 

Name  all  the  pairs  of  factors  that  will  produce  36  ?  860  ? 
48?480?  60?  600?  64?  640  ?  72  ?  720  ? 

*Note  4.  '   . 


24  MENTAL   ARITHMETIC. 

31.  What  is  one  ninth  of  27  ?  A  man  divided  270  dollars 
equally  among  9  persons ;  how  much  did  he  give  to  each  ? 

32.  What  is  one  fourth  of  48  ?  If  480  dollars  are  divided 
into  4  equal  shares,  what  will  each  share  be  ? 

What  is  one  eighth  of  480  ?  one  sixth  of  480  ?  one  twelfth 
of  480? 

33.  What  is  one  seventh  of  63  ?  If  a  ship  sails  at  a  uni- 
form rate,  630  miles  in  a  week,  how  many  miles  does  she  sail 
in  a  day  ? 

What  is  one  ninth  of  630  ?  What  is  one  sixth  of  630  ? 
What  is  one  third  of  630  ? 

34.  What  is  one  fifth  of  25  ?  If  250  trees  are  placed  in  5 
equal  rows,  how  many  will  there  be  in  each  row  ? 

If  placed  in  50  equal  rows,  how  many  would  there  be  in 
each  row  ? 

35.  What  is  one  fourth  of  36?  In  a  circle  there  are  360 
degrees ;  how  many  are  there  in  one.  fourth  of  a  circle  ?  How 
many  in  one  eighth  of  a  circle  ?  How  many  in  one  sixteenth 
of  a  circle  ? 

36.  What  is  one  eighth  of  56  ?  If  560  trees  were  planted 
in  8  equal  rows,  how  many  would  there  be  in  each  row  ?  If 
planted  in  1 6  rows,  how  many  would  there  be  in  each  row  ? 

37.  What  is  one  eleventh  of  55  ?  If  you  place  550  trees  in 
11  equal  rows,  how  many  will  there  be  in  a  row?  If  you 
place  them  in  fifty  rows,  how  many  will  there  be  in  each 
row  ?  If  you  place  them  in  25  rows,  tow  many  will  there  be 
in  each  row? 

38.  What  is  one  twelfth  of  96  ?  If  a  man  spends  960  dol- 
lars in  a  year,  how  much  will  be  his  average  expense  for 
each  month? 

39.  What  is  one  tenth  of  40  ?  of  400  ?  of  4000  ? 

What  is  one  fourth  of  40  ?  400  ?  of  4000  ?  of  80  ?  of  800  ? 
of  8000  ?     One  fourth  of  12  ?  of  120  ?  of  1200  ?  of  12,000. 

40.  What  is  one  fifteenth  of  60  ?  of  600  ?  of  6000  ? 
What  is  one  thirtieth  of  60  ?  of  600  ?  of  6000  ?  of  1200  ? 

of  12,000? 

41.  What  is  one  fifth  of  92?  What  is  one  third  of  51  ? 
One  fourth  of  65  ?  One  fifth  of  78?  One  sixth  of  96  ?  One 
seventh  of  100?  Divide  into  70  and  30.  What  is  one 
ninth  of  117  ?  Ajis.  One  ninth  of  90  is  10  ;  one  ninth  of  27 
is  3  ;  10  and  3  are  13. 

42.  What  is  one  third  of  49  ?  one  sixth  of  84  ?  one  fifth 
of  79  ?  one  eighth  of  100  ?  one  seventh  of  91  ?  one  sixth  of 


DIVISION.  S5 

79?  one  fourth  of  76 ?  of  92?  of  57?  of  60?  of  52?  of  65? 
of  70? 

43.  What  is  one  fourth  of  480  ?  What  is  one  fifth  of  155  ? 
Divide  into  150  and  5.  What  is  one  fourth  of  920  ?  What 
is  one  fifth  of  15,76*5.  This  number  may  be  divided  into 
15,000,  750  and  15,  or  15,000,  500,  250  and  15. 

44*  What  is  one  sixth  of  4836?  One  eighth  of  336? 
Divide  into  320  and  16.  What  is  one  seventh  of  574?  one 
third  of  684  ?  one  sixth  of  43,248  ?  one  ninth  of  72,108  ?  of 
64,827  ?  one  fifth  of  5275  ?  one  fourth  of  92,648  ?  , 

r  45.  What  is  one  third  of  6156  ?  of  8436  ? 

46.  What  is  one  fourth  of  6428  ?  of  9648  ? 

47.  What  is  one  fifth  of  7655  ?  of  12,535  ? 

48.  What  is  one  sixth  of  13,218  ?  of  1944^. 

49.  What  is  one  seventh  of  10,542  ?  of  14,280  ? 

50.  What  is  one  eighth  of  1632  ?  of  2560  ?  fy-C^ 

SECTION    V. 

TABLE   OF  TIME. 

60  Seconds,  [sec]   make 1  Minute,     marked     m. 

60  Minutes 1  Hour, h. 

24  Hours 1  Day, d. 

7  Days    1  Week, w. 

4  Weeks 1  Month, mo. 

52  Weeks,  1  day,  6  hours 1  Year, y. 

365  Days,  6  Hours 1  Year,  - y. 

12  Calendar  Months 1  Year, y. 

In  common  reckoning,  4  weeks  are  called  a  month,  but  this 
is  merely  for  convenience  in  doing  business.  The  number  of 
days  in  a  calendar  month  is  30  or  31 ;  except  February, 
which  has  28  days,  and  in  leap  year  29.  The  6  hours  over 
and  above  the  365  days  in  a  year,  will  in  4  years  amount  to 
a  whole  day ;  it  is  then  added  to  February,  making  29  days, 
and  that  year  is  called  leap  year.     The  number  of  days  in 

*  Note  5. 


26  MENTAL    ARITHMETIC. 

the  Other  months  may  be  seen  in  the  line  below.  The  months 
connected  by  a  tie  drawn  over  the  words  have  31  days  ;  those 
connected  by  a  tie  underneath  have  30.  V 


Jan.  Feb.  March.  April.  May.  June.  July.  August.  Sept.  Oct.  Nov.  Dec. 
28. 29.  ^ '  > ' 

You  observe  that  beginning  with  January,  every  alternate 
month  has  31  days,  till  you  come  to  July  and  August.  Here 
there  are  two  months  together  that  have  31,  and  then  the  al- 
ternation goes  on  as  before  to  the  end  of  the  year.  * 

The  leap  year  may  be  easily  known  from  the  fact  that  the 
number  of  the  year  is  exactly  divisible  by  4.  Thus  1844 
was  leap  year ;  the  number  can  be  divided  by  4. 

What  years  in  the  present  century  have  been  leap  years  ? 
What  years  will  be  leap  years  from  now  to  the  close  of  the 
century*? 

•  1.  ia  onp  mmut§  t6e|;e'^re  60  seconds  ;  in  one  hour  there 
are  60  minutes.  iTo^  niar^y  s^c^ds  af^^  there  in  one  hour  ? 
How  many  in  10  hours  ?  How  man/  in  20  h»urs  ?  How 
many  in  one  day  ?  How  many  in  seven  days,  or  one  week  ? 
How  many  in  ten  days  ?  In  100  days  ?  In  300  days  ?  In 
350  days?     In  365  days? 

2.  If  you  save  30  minutes  from  idleness  each  day,  how 
many  hours  will  you  save  in  a  week  ?  How  many  in  5 
wfeeks  ?    How  many  in  50  weeks  ?   How  many  in  52  weeks  ? 

3.  If  you  read  40  pages  each  day,  how  many  pages  will  you 
read  in  one  week  ?  How  many  in  10  weeks  ?  How  many 
in  52  weeks  ? 

4.  If  a  printer  sets  4  pages  of  type  in  a  day,  in  how  many 
days  will  he  set  the  type  for  a  book  of  500  pages  ?  What 
will  his  wages  come  to  at  Sl,50  a  day  ? 

5.  If  there  are  300  members  in  the  Legislature  of  Massa- 
chusetts, and  each  member  receives  2  dollars  a  day  during 
the  session,  what  does  the  pay  of  all  the  members  come  to  for 
one  day  ?  What  does  the  pay  of  the  Legislature  amount  to 
for  one  week  ?     For  10  weeks  ? 

6.  The  number  of  members  in  Congress  is  about  275.  At 
8  dollars  a  day,  what  is  the  amount  of  their  pay  each  day  ? 
Wliat  would  be  the  amount  of  their  pay  for  10  days  ?  For 
100  days  ? 

7.  How  many  days  are  there  in  the  3  months  of  spring  ? 


TIME.  27 

How  many  days  in  the  3  months  of  summer  ?     How  many 
days  in  autuimi  ? 

8.  How  many  days  in  the  winter  of  leap  year  ?  How 
many  days  were  there  in  the  winter  of  1844  ?  How  many 
days  in  the  winter  of  1845  ? 

9.  If  January  comes  in  on  Monday,  on  what  day  of  the 
week  will  February  come  in  ? 

If  March  comes  in  on  Wednesday,  on  what  day  of  the 
week  will  April  come  in  ? 

If  August  comes  in  on  Saturday,  on  what  day  of  the  week 
will  September  come  in  ? 

10.  If  April  comes  in  on  Sunday,  on  what  day  of  the  week 
will  it  go  out  ? 

If  June  comes  in  on  Tuesday,  on  what  day  will  it  go  out  ? 
If  September  comes  in  on  Saturday,  on  what  day  will  it  go 
out? 

11.  If  January  comes  in  on  Friday,  how  many  Sundays 
will  there  be  in  that  month  ? 

If  it  comes  in  on  Thursday,  how  many  Sundays  will  there 
be  in  that  month  ? 

If  June  comes  in  on  Friday,  how  many  Sundays  will  there 
be  in  that  month  ?     If  it  comes  in  on  Saturday,  how  many  ? 

If  February  comes  in  on  Saturday,  and  that  year  is  leap 
year,  how  many  Saturdays  will  there  be  in  the  month  ?  If 
it  is  not  leap  year,  how  many  ? 

In  1845  February  came  in  on  Saturday,  how  many  Sat- 
urdays were  there  in  that  month  ? 

In  1844  February  came  in  on  Thursday,  how  many  Thurs- 
days were  there  in  that  month  ? 

12.  If  January  comes  in  on  Monday,  on  what  day  of  the 
week  will  March  come  in,  if  it  is  leap  year  ?  On  what  day, 
if  it  is  not  leap  year  ? 

13.  K  June  comes  in  on  Wednesday,  what  day  of  the  weei 
will  the  1st  of  August  be  ?     The  9th  ?  the  12th  ?  the  15th  ? 


28  MENTAL    ARITHMETIC. 

TABLE  OF  LINEAR  MEASURE. 

12  inches, 1  foot, ft. 

3  feet, 1  yard,  • yd. 

5J  yards,  16J  feet, 1  rod, rd. 

40  rods, 1  furlong, •  •  fur. 

8  furlongs=;320  rods, 1  mile, m. 

3  miles, 1  league, 1. 

69  J  miles, 1  degree  of  latitude,  •  •  •  •  deg. 

For  lengths  less  than  an  inch,  the  inch  is  divided  into 
fourths,  eighths,  tenths,  or  twelfths. 

1.  How  many  inches  in  2  feet?  In  4  feet?  In  5  feet? 
In  7  feet  ?  In  10  feet  ?  In  12  feet  ?  How  many  inches  in 
4  yards  ?  In  1  rod  ?  In  3  rods  ?  How  many  feet  in  1  fur- 
long ?     In  2  furlongs  ?     In  4  furlongs  ?    In  1  mile  ? 

2.  How  many  miles  in  46  leagues  ?  In  132  leagues  ? 
How  many  miles  in  2  degrees  of  latitude  ?  In  3  degrees  ? 
In  4^  degrees  ?     In  6  degrees  ? 

In  estimating  the  miles  in  any  number  of  degrees  of  latitude, 
it  is  most  convenient  to  call  a  degree  70  miles,  and  then,  if 
we  wish  to  be  accurate,  we  may  subtract  from  the  answer  half 
as  many  miles  as  there  are  degrees.  In  this  way  the  distance 
of  places  from  each  other  may  be  determined  on  a  map :  the 
degrees  of  latitude  on  the  margin  may  be  used  as  a  scale  of 
miles.  If  the  distance  of  two  places  from  each  other  is  equal 
to  6|  degrees  of  latitude,  how  many  miles  are  they  apart  ? 

3.  How  many  yards  in  10  rods  ?  In  20  rods  ?  In  30  rods  ? 
In  1  furlong  ?     In  8  furlongs,  or  1  mile  ? 

4.  In  measuring  land  or  a  road  with  a  chain  4  rods  long, 
how  many  times  must  the  chain  be  applied  to  the  ground  in 
measuring  one  mile  ?  How  many  times  in  measuring  the  road 
from  Boston  to  Salem,  15  miles  ?  How  many  in  measuring 
from  Boston  to  Providence,  40  miles  ? 

5.  If  a  man  walks  three  miles  in  an  hour,  how  many  min- 
utes will  he  be  in  walking  1  mile  ?  How  many  minutes  in 
walking  1  fourth  of  a  mile  ?  How  many  rods  will  he  walk 
in  1  minute?     Ans.  16. 

How  many  seconds  will  he  be,  then,  in  walking  1  rod  ? 
16  will  go  into  60,  3  times  and  12  over.  He  will  be,  then,  a 
little  less  than  4  seconds  in  walking  1  rod. 

Let  us  now  suppose  he  is  precisely  4  seconds  in  walking 


REDUCTION   OF   LINEAK   MEASUHE.  29 

1  rod  ;  how  many  rods  would  he  walk  in  a  minute  ?  How 
many  in  10  minutes?  How  many  in  60  minutes?  How 
many  miles  ? 

6.  If  a  man  in  walking  takes  6  steps  to  a  rod,  how  many 
steps  will  he  take  in  walking  a  mile  ?  How  many  in  walking 
10  miles  ?     How  many  in  walking  40  miles  ? 

7.  If  a  man  in  walking  takes  6  steps  to  a  rod,  and  takes  2 
steps  in  a  second,  how  many  seconds  will  he  be  in  walking 
one  rod  ?  How  many  seconds  in  walking  10  rods  ?  20  rods  ? 
If  a  man  walks  20  rods  in  one  minute,  how  many  minutes 
will  it  take  him  to  walk  a  mile  ?  20  are  contained  in  320  just 
as  many  times  as  2  are  contained  in  32. 

8.  If  a  man  walks  20  rods  in  one  minute,  how  long  will  it 
take  him  to  walk  4  miles  ? 

9.  If  a  rail-road  train  goes  30  miles  in  an  hour,  how  faf 
does  it  go  in  one  minute  ?     How  many  rods  in  one  second  ? 

Ans.  30  miles  in  60  minutes  is  1  mile  in  2  minutes ;  half  a 
mile  in  one  minute  ;  quarter  of  a  mile  in  half  a  minute  ;  that 
is  80  rods  in  30  seconds ;  that  is  8  rods  in  3  seconds ;  and  in  1 
second,  one  third  of  8  rods  or  2  rods  and  two  thirds. 

10.  How  many  rods  in  14  miles  ? 

In  one  rod  there  are  16^^  feet.  In  one  mile  there  are  320 
rods ;  how  many  feet  are  there  in  a  mile  ?  There  are  various 
ways  of  finding  the  answer  to'this  question  ;  some  of  them  will 
be  suggested,  and  the  pupil  left  to  take  his  choice. 

First,  how  many  feet  are  there  in  300  rods  ? 

This  is  not  difficult,  for  in  3  rods  there  are  three  times  16 J 
feet,  and  in  300  rods  there  100  times  as  many.  3  times  15 
feet  are  45  feet ;  3  times  1^  feet  are  4^,  which  added  to  45 
make  49  J  feet  in  3  rods.  Now  100  times  49  are  4900,  and 
100  halves  are  50 ;  4950  feet  in  300  rods.  In  20  rods,  there 
are  ten  times  as  many  feet  as  in  2  rods ;  in  2  rods  there  are 
twice  16^  or  33 ;  in  20  rods  therefore  there  are  330  feet;  300 
added  to  4900  make  5200,  and  30  added  to  50  make  80 ; 
there  are  then  5280  feet  in  a  mile. 

Another  method  would  be  to  multiply  320  first  by  8,  and 
that  product  by  2,  for  8  and  2  are  the  factors  of  16  ;  then  as 
there  was  ^  a  foot  in  each  rod  left  out,  there  must  be  added 
half  as  many  feet  as  there  are  rods,  or  half  of  320. 

Another  method  would  be  to  mtiltiply  320  by  10,  then  by 
six  and  add  the  products,  and  lastly  by  J  and  add  that  to  the 
other  products. 

3* 


30  MENTAL    ARITHMETIC. 

The  pupil  can  tiy  each  of  these  ways,  and  see  if  he  obtains 
the  same  answer. 

Let  us  now  see  if  our  answer  is  correct.  If  there  are  5280 
feet  in  a  mile,  how  many  are  there  in  half  a  mile  ?  One  half 
of  5200  is  2600,  one  half  of  80  is  40 ;  there  are  then  2640 
feet  in  half  a  mile.  How  many  in  1  fourth  of  a  mile  ?  One 
half  of  2640  feet,  which  is  1320.  Now  1  fourth  of  a  mile  is 
80  rods.  If  then  there  are  1320  feet  in  80  rods,  how  many 
will  there  be  in  8  rods  ?  One  tenth  as  many.  One  tenth  of 
1320  is  132.  Now  how  many  are  there  in  one  rod  ?  One 
eighth  of  132 :  dividing  132  into  80  and  52 ;  one  eighth  of 
80  is  10,  and  one  eighth  of  52  is  6|  or  J,  which  added  to  10 
make  16|.  We  have  now  come  down  from  5280,  and  arrived 
by  successive  divisions  to  161^,  the  number  from  which  we 
started  at  first.     The  answer  is  thus  proved  to  be  correct. 

11.  How  many  feet  are  there  in  2  rods  ?  In  20  rods  ?  In 
200  rods  ? 

12.  How  many  feet  are  there  in  3  rods  ?  In  30  rods  ?  In 
300  rods  ?     In  8  rods  ?     In  80  rods,  or  quarter  of  a  mile  ? 

13.  How  many  rods  are  there  in  2  miles  ?  In  4  miles  ? 
In  8  miles  ?     In  20  miles  ?     In  30  miles  ?     In  50  miles  ? 

14.  How  many  rods  are  there  in  half  a  mile  ?  In  three 
fourths  of  a  mile  ?  In  one  mile  and  a  half?  In  one  mile  and 
three  furlongs  ?  In  two  miles  and  five  furlongs  ?  In  4  miles 
and  7  furlongs  ? 

15.  How  many  yards  are  there  in  2  rods  ?  In  20  rods  ? 
In  3  rods  ?     In  30  rods  ?     In  300  rods  ? 

How  many  yards  are  there  in  3  rods  and  4  feet  ?  How 
many  yards  are  there  in  17  rods  11  feet? 

16.  How  many  inches  are  there  in  7  feet  ?  In  9  feet  ?  In 
6  feet  ?     In  8  feet  and  6  inches  ?     In  11  feet  9  inches  ? 

17.  How  many  inches  are  there  in  1  rod,  or  16 J  feet? 
How  many  inches  in  2  rods  ?     In  3  rods  ?     In  4  rods  ? 

18.  A  house  is  46  feet  and  5  inches  in  length,  how  many 
inches  long  is  it  ? 

A  creeping  vine  grows  on  an  average  3  inches  a  day ;  how 
many  days  will  it  take  to  grow  from  the  ground  to  the  top  of 
a  house  that  is  25  feet  high  ? 

19.  A  stage-horse  travels  13  miles  and  "20  rods  each  day; 
how  far  will  he  travel  in  6f)  days  ?     How  far  in  120  ? 

20.  How  far  will  the  horse  travel  in  a  year,  if  he  rests  5 
days  in  the  year  ? 


VELOCITY    OF   SOUND.  81 

21.  What  is  the  weight  of  iron  used  in  one  mile  of  rail- 
road, allowing  55  pounds  for  a  yard  of  rail  ? 

One  yard  of  heavy  rail  weighs  55  pounds.  Twice  this,  or 
110  pounds,  would  be  the  weight  of  both  rails  for  one  yard 
of  a  single  track.  Five  and  a  half-times  this  would  be  the 
weight  for  one  rod.  From  this  may  be  obtained  the  weight 
for  10  rods  ;  for  100  rods  ;  for  300  rods  ;  for  320  rods. 

22.  What  would  be  the  cost  of  the  iron  for  a  single  track 
of  one  mile  of  rail-road  at  4  cents  a  pound?  How  much 
would  be  saved  in  the  expense  of  the  iron  for  one  mile  of  rail- 
road, if  the  price  of  iron  should  be  reduced  one  cent  a  pound  ? 

23.  If  the  cost  of  the  iron  for  a  single  track  of  rail-road  is 
6000  dollars  a  mile,  and  the  cost  of  the  land  and  the  labor  of 
construction  equals  that  of  the  iron,  what  would  be  the  cost 
of  15  miles  of  rail-road  ?  of  24  miles  ? 

24.  What  would  be  the  cost  of  constructing  one  mile  of 
common  road  at  $2,25  a  rod  ? 

25.  What  would  be  the  cost  of  building  80  rods  of  common 
wall  at  54  cents  a  rod  ? 

26.  If  a  horse  travels  10  miles  in  an  hour,  how  long  is  he 
in  traveling  1  mile  ?  How  long  in  traveling  J  of  a  mile,  or 
80  rods  ?  How  many  seconds  is  he  in  traveling  8  rods  ? 
How  long  in  traveling  1  rod  ? 

27.  A  body  falling  -through  the  air,  falls  in  the  first  second 
16  J  feet,  and  each  succeeding  second  it  falls  twice  16  J  feet 
further  than  in  the  preceding  second.  How  far  would  a  stone 
fall  in  2  seconds  ? 

28.  How  far  would  it  fall  in  the  third  second  ?  How  far 
would  it  fall  in  3  seconds  ? 

29.  How  far  would  it  fall  in  the  fourth  second  ?  How  far 
would  it  fall  in  4  seconds  ? 

30. -Sound  moves  through  the  air  at  the  rate  of  1090*  feet 
in  a  second.  How  many  feet  will  it  move  in  3  seconds  ?  How 
many  feet  in  4  seconds  ?     How  many  feet  in  5  seconds  ? 

As  sound  is  found  thus  to  pass  5450  feet  in  5  seconds,  and 
as  there  are  5280  feet  in  a  mile,  we  see  that  in  5  seconds 
sound  moves  170  feet  more  than  a  mile.  Now  as  165  feet  is 
just  10  rods,  we  say,  without  much  error,  that  sound  moves 
1  mile  and  10  rodsAn  5  seconds.  This  is  accurate  enough  for 
all  common  purposes,  and  you  will  do  well  to  fix  it  in  your 
memory,  and  make  your  calculations  from  it. 

*  Professor  Pierce  on  Soands. 


32  MENTAL    ARITHMETIC. 

81.  How  many  rods  will  sound  move  in  1  second?  One 
fifth  of  320+10  rods,  =  66  rods. 

32.  How  many  rods  in  2  seconds  ?  How  many  rods  in  4 
seconds  ? 

Thus,  if  you  watch  the  stroke  of  an  axe  used  by  some  one 
at  a  distance,  and  observe  that  the  sound  comes  to  you  one 
second  later  than  you  see  the  stroke,  you  may  know  that  the 
distance  is  66  rods.  If  the  sound  of  a  bell  comes  to  you  two 
seconds  after  the  stroke  is  given,  you  must  be  distant  from  the 
bell  132  rods.  In  these  cases  no  allowance  is  made  for  the 
transmission  of  light.  You  are  supposed  to  see  the  motion  as 
soon  as  it  occurs.  This  is  not  strictly  the  fact ;  but  the  time 
is  so  exceedingly  small  that  it  need  not  be  taken  into  the 
account. 

33.  In  a  still  night  a  church  bell  is  sometimes  heard  at  the 
distance  of  12  miles ;  how  many  seconds,  or  nearly  how  many, 
after  the  stroke  would  the  sound  be  heard  at  that  distance  ? 

34.  If  the  report  accompanying  a  flash  of  lightning  is  heard 
4  seconds  after  the  flash  is  seen,  how  far  from  the  hearer  was 
the  discharge  ?  How  far,  if  the  time  between  the  flash  and 
the  report  is  6  seconds  ?  How  far,  if  the  time  is  8  seconds  ? 
How  far,  if  the  time  is  10  seconds  ?  How  far,  if  the  time  is 
15  seconds  ? 

35.  The  report  of  a  cannon  has,  in  some  instances,  been 
heard  at  the  distance  of  100  miles :  in  how  many  seconds,  or 
nearly  how  many,  after  the  discharge,  would  the  report  be 
heard  at  that  distance  ?     In  how  many  minutes  ? 

36.  By  means  of  a  magnetic  telegraph  it  is  possible  to  com- 
municate intelligence  instantly  from  New  Orleans  to  Boston,  a 
distance  of  1500  miles.  If  this  intelligence  could  be  commu- 
nicated by  sound  passing  through  the  air,  how  long  would  it 
be  traveling  that  distance,  allowing  5  seconds  to  a  mile  ? 

A  ball  discharged  from  a  gun  moves  at  first  with  a 
greater  speed  than  sound,  but  it  moves  slower  and  slower, 
and  before  it  is  spent  the  report  overtakes  it,  and  passes  by 
it :  for  sound  moves  always  at  the  same  rate. 

37.  If  a  cannon  ball  moves  a  mile  in  8  seconds,  how  long 
would  it  be  in  moving  3  miles  ?  How  long  in  moving  one 
fourth  of  a  mile  ?  How  long  in  moving  one  eighth  of  a  mile  ? 
How  long  in  moving  If  miles  F 


REDUCTION, MONEY.  83 

SECTION    VI. 

TABLE  OF  FEDERAL  MONEY. 

10  IVIills  make  •  •  •  • 1  Cent, marked  •  • .  •  ct. 

10  Cents, 1  Dime, d. 

10  Dimes, 1  Dollar, D. 

10  DoUars, 1  Eagle, E. 

This  is  established  by  law  as  the  currency  of  the  United 
States. 

The  general  mark  for  Federal  Money  is  $,  as  $5.14,  five 
dollars  fourteen  cents.  A  period  must  always  be  placed  be- 
tween dollars  and  cents. 

1.  How  many  mills  in  2  cents  ?  In  10  cents  ?  In  12 
cents?  In  5J  cents?  In  12J  cents?  In  36  cents?  In  1 
dollar? 

2.  How  many  cents  in  5  dimes?  In  11  dimes?  In  16 
dimes  ?     In  4^  dollars  ?     In  17^  dollars  ?     In  12|  dollars  ? 

3.  How  many  dimes  in  7  dollars  ?  In  13  J  dollars  ?  In  3 
eagles  ?     In  56  dollars  ?     In  100  dollars  ? 

4.  How  many  cents  in  35  mills  ?  In  180  mills  ?  In  600 
mills  ?  How  many  dimes  in  80  cents  ?  In  210  cents  ?  In 
740  cents  ? 

5.  How  many  dollars  in  350  cents  ?  In  325  cents  ?  In 
700  cents  ?  In  850  cents  ?  In  1400  cents  ?  In  1675  cents  ? 
In  925  cents  ? 

TABLE  OF  STERLING  MONEY. 

4  farthings  [qr.]  make*  •  •!  penny, marked*  'd. 

12  pence,  • 1  shilling, s. 

20  shillings, 1  pound, £. 

This  is  the  currency  of  Great  Britain. 

1.  How  many  farthings  are  there  in  3  pence  ?  In  7  pence  ? 
In  8  pence  ?     In  10  pence  ?     In  11  pence  ? 

2.  How  many  pence  in  2  shillings?  In  12  shillings?  In 
15  shillings  ?     In  18  shillings  ?     In  16  shillings  ? 

3.  How  many  shillings  in  4  pounds  ?  In  7  pounds  ?  In 
18  pounds  ?     In  36  pounds  ?     In  84  pounds  ? 


34  MENTAL    ARITHMETIC. 

4.  How  many  farthings  in  1  shilling  and  6  pence  ?  In  2 
shillings  and  6  pence  ?     In  15  shillings  and  4  pence  ? 

How  many  pence  in  10  shillings  ?  In  20  shillings  ?  In  2 
pounds  ?     In  4  pounds  ?     In  12  pounds  ? 

5.  How  many  farthings  in  1  pound  ?  In  5  pounds  ?  In 
8  pounds  ?     In  1  pound  2  shillings  ? 

6.  How  many  pence  in  45  farthings  ?  In  128  farthings  ? 
In  464  farthings  ?     In  1296  farthings  ?     In  648  farthings  ? 

7.  How  many  shillings  in  80  pence  ?  In  67  pence  ?  In 
372  pence  ?     In  649  pence  ?     In  840  pence  ? 

8.  How  many  pounds  in  267  shillings  ?  In  845  shillings? 
In  432  shiUings  ?     In  640  shillings  ?     In  4000  shillings  ? 

9.  How  many  pounds  in  890  pence  ?  In  16,000  farthings  ? 
In  720  pence  ?     In  1200  pence  ?     In  456  pence  ? 

10.  How  many  pence  in  5  pounds  4  shillings  ?  In  7 
pounds  8  shillings  ?     In  12  pounds  3  shillings  ? 

How  many  farthings  in  4  shillings^pence  ?  In  9  shilhngs  ? 
How  many  farthings  in  6  pounds  3  shillings  8  pence  ? 

11.  A  man  set  out  on  a  journey  with  £4  8s  6d  in  his 
pocket :  before  spending  any  thing,  he  received  in  payment  of 
a  debt  £2  3s  8d.  How  much  had  he  then  ?  When  he  arrived 
home  he  had  spent  £1  4s  6d.     How  much  had  he  then  ? 

These  denominations,  you  must  bear  in  mind,  have  not  the 
same  value  in  English  currency,  that  they  have  in  the  United 
States. 

In  our  country  they  have  different  values  in  the  different 
States,  but  in  none  of  them  so  high  a  value  as  in  England. 
In  the  New  England  States  a  shilling  is  equal  to  1 6  cents  and 
two  thirds, -and  6  shillings  make  a  dollar.  In  New  York  12 
and  a  half  cents  are  a  shilling,  and  8  shillings  a  dollar.  In 
other  States  the  values  are  still  different ;  but  these  denomi- 
nations are  gradually  giving  way  to  those  of  the  Federal 
currency.  They  are  now  used  only  in  naming  prices.  Ac- 
counts are  not  kept  in  them,  and  all  that  is  important  in  them 
may  be  learned  by  practice  without  further  notice  here. 

In  the  Sterling  currency,  used  in  England,  a  pound  is  equal 
to  4  dollars,  44  cents  and  4  mills ;  10  shillings,  therefore,  or 
half  a  pound,  are  2  dollars,  22  cents,  2  mills ;  and  1  shilling  is 
one  tenth  part  of  that,  or  22  cents,  2  mills.  An  English  six- 
pence is,  therefore,  11  cents  1  mill.  The  following  table  will 
be  useful  in  exchanging  English  money  to  our  own. 


REDUCTION.  35 

1  pound,  £,  is $4.44  4 

10  shillings,  or  half  a  pound, 2.22  2 

1  shilUng, 22  2 

6d,  or  half  a  shilling, Ill 

4  shillings,  6  pence, 1.00  0 

1  guinea,  21  shillings, 4.66  6 

The  actual  value  of  the  English  money  is  a  little  higher 
than  is  here  stated,  but  this  is  sufficiently  accurate  for  a  gen- 
eral table. 

1.  What  is  the  value  in  dollars  and  cents  of  2£  ?  3£  ?  4£  ? 
5£?  l£6s?  2£8s?  3s  6d?  5s  9d? 


TABLE  OF  DRY  MEASURE. 

2  pints,  [pt.]  make •  1  quart, marked  qt. 

4  quaa-ts,  •  •  ^   1  gallon, gal. 

8  quarts, 1  peck, pk. 

4  pecks, 1  bushel, bu. 

8  bushels, 1  quarter, qr. 

36  bushels, 1  chaldron, ch. 

These  denominations  are  used  for  measuring  grain,  fruit, 
and  coal.  The  pint,  quart,  and  gallon  are  larger  than  the 
same  denominations  in  wine  measure,  and  less  than  those  of 
beer  measure. 

1.  How  many  pints  in  1  peck?  In  3  pecks?  In  1  bushel? 
In  3  bushels  ?     In  4  bushels  ? 

2.  How  many  quarts  are  there  in  1  bushel  ?  In  4  bushels  ? 
How  many  pecks  in  7  quarters  ?     In  2  chaldrons  ? 

3.  If  a  horse  eat  4  quarts  of  oats  each  day,  how  many 
bushels  will  he  eat  in  10  weeks  ?  How  many  bushels  in  50 
weeks  ?     In  52  weeks  ? 

What  will  they  cost  at  50  cents  a  bushel  ? 

4.  In  80  quarts  how  many  pecks  ?     How  many  bushels  ? 
In  644  quarts  how  many  pecks  ?     How  many  bushels  ? 
In  7840  quarts  how  many  pecks  ?     How  many  bushels  ? 

5.  In  100  pints  how  many  pecks  ?     How  many  bushels  ? 
In  620  pints  how  many  pecks  ?     How  many  bushels  ? 


36  MENTAL    ARITHMETIC. 

TABLE  OF  AVOIRDUPOIS  WEIGHT. 

16  drams,  [dr.]  make 1  ounce, marked  oz. 

16  ounces,  •  • 1  pound, lb. 

25  pounds, 1  quarter, (net  wt.)  qr. 

28  pounds, 1  quarter,  •  •  •  (gross  wt.)  qr. 

4  quarters, 1  hundred  weight, cwt. 

20  hundred  weight, 1  Ton, T. 

,  These  denominations  are  used  in  weighing  hay,  grain,  meat, 
flour,  and  all  the  most  common  articles  bought  and  sold  by 
weight.  On  account  of  the  waste  in  handling  such  articles, 
their  shrinking  in  drying,  and  worthless  admixtures  sometimes 
found  in  them,  112  pounds  are  sometimes  allowed  for  one 
hundred  weight ;  this  makes  28  pounds  one  quarter,  and  is 
called  gross  w^eight.  In  all  the  following  questions  of  Avoir- 
•dupois  wt.,  understand  gross  wt.,  unless  net  wt.  is  expressed. 

1.  How  many  drams  in  3  oz. ?  In  5  oz. ?  In  8  oz.?  In 
11  oz. .?  How  many  oz.  in  12  lbs.?  In  15  lbs.?  In  20  lbs.? 
In  32  lbs.?     In  45  lbs.? 

2.  How  many  lbs.  in  4  cwt.  net  weight  ?  In  4  cwt.  gross  ? 
In  6  cwt.  net  weight  ?  In  6  cwt.  gross  ?  In  5  cwt.  2  qrs. 
net  ?  In  5  cwt.  2  qrs.  gross  ?  In  7  cwt.  3  qrs.  net  weight  ? 
In  7  cwt.  3  qrs.  gross  ? 

37  How  many  lbs.  in  a  ton,  net  weight  ?  In  a  ton,  gross  ? 
How  many  lbs.  in  5  tons,  3  cwt,  net  ?    In  5  tons,  3  cwt.  gross  ? 

4.  There  are  2  loads  of  hay  whose  net  weight  is  as  follows ; 
the  first,  25  cwt.  3  qrs.  17  lbs. ;  the  second,  17  cwt.  2  qrs.  21 
lbs.     "What  is  the  weight  of  both  ? 

5.  A  man  set  out  for  market  with  a  load  of  hay  weighing 
36  cwt.  2  qrs.  15  lbs.,  net  weight ;  he  lost  a  part  of  it ;  the  re- 
mainder weighed  25  cwt.  1  qr.  8  lbs.    How  much  did  he  lose  ? 

6.  If  there  are  196  lbs.  in  a  barrel  of  flour,  how  many 
pounds  net  weight  are  there  in  10  barrels  ? 

196  lbs.  are  7  quarters  gross ;  how  many  cwt.  gross  are  there 
in  10  barrels  of  flour  ? 

7.  How  many  pounds  are  there  in  100  oz.?     In  650  oz.? 

8.  A  barrel  of  flour  weighs  7  quarters  gvoss ;  how  many 
tons  gross,  are  there  in  100  barrels  of  flour  ? 

9.  What  will  be  the  expense  of  transporting  by  rail-road 
100  bai*rels  flour,  100  miles,  at  the  rate  of  3  dollars  a  ton  ? 

What  will  be  the  expense  of  transporting  a  single  barrel  ? 


REDUCTIO¥.  37 

100  barrels  are  700  qrs.  gross  weight,  400  qrs.=100  cwt= 
5  tons  :  300  qrs.=75  cwt.=3  tons,  15  cwt. ;  this  added  to  5 
tons,  makes  8  tons,  15  cwt. 

10.  The  freight  of  goods  by  wagon  is  about  20  dollars  a 
ton' gross  for  100  miles ;  at  this  rate  what  will  be  the  cost  of 
carrying  a  barrel  of  flour  100  miles  ? 


TABLE  OF  TROY  WEIGHT. 

"    24  grains  [gr.]  make 1  pennyweight,    dwt. 

20  pennyweights, 1  ounce,   oz. 

12  ounces, 1  pound,   lb. 

This  is  used  for  weighing  gold  and  silver.  The  pound 
Troy  is  nearly  one  fifth  less  than  the  pound  Avoirdupois. 

1.  How  many  grains  in  6  pennyweights  ?  In  8  penny- 
weights? In  12  pennyweights?  Inloz. ?  In2oz. ?  In 
4  oz.  ?     In  6  oz.  ? 

2.  How  many  pennyweights  in  8  oz.  ?  In  11  oz.?  In  1 
lb.?  In  3  lbs.?  In  8  lbs.?  In  5  lbs.  ?  Inllb.3oz.?  In 
Slbs.Soz.? 

3.  How  many  oz.  in  120  dwt.?  In  480  dwt.?  In  960 
grs.  ?  How  many  lbs.  in  100  oz.  ?  In  860  dwt.  ?  In  1200 
dwt.? 

TABLE  OP  APOTHECARIES'  WEIGHT. 

20  grs.  make  •  •  •    • 1  scruple,  marked 9 

3  scruples, 1  dram, 5 

8  drams, 1  ounce,    S 

12  ounces,    1  pound, ib 

This  table  is  used  only  by  apothecaries  in  mixing  medi- 
cines.    The  pound  and  ounce  are  the  same  as  in  Troy  weight. 

TABLE   OF  CLOTH  MEASURE. 

2^  inches  [in.]  make  •  •  •  •  1  nail,  marked na. 

4    nails, 1  quarter, qr. 

4  quarters, 1  yard, yd. 

3    quarters, 1  ell  Flemish, Fl.  e. 

5  quarters, 1  ell  English, E.  e. 

6  quarters, 1  ell  French,   Fr.  e. 


38  MENTAL   ARITHMETIC. 

1.  How  many  inches  in  1  qr.  ?     In  1  yd.?     In  3  yds. ?  In 
1  EU.  Eng.  ?     In  1  Ell.  Fr.  ?     In  1  Ell.  Fl.  ? 

2.  How  many  inches  in  4  yds.  ?     In  7  yds.  ?     In  12  yds.  ? 
In  10  yds.  ?    In  20  yds.  ?    In  6  yds.  3  qrs.  ?    In  4  yds.  1  qr.  ? 


TABLE  OF  WINE  MEASURE. 

4  gUls  [gi.]  make*  • 1  pint,  marked pt. 

2  pints, 1  quart, qt. 

4  quarts, 1  gallon, gal. 

^1^  gallons, 1  barrel, bl. 

63  gallons, 1  hogshead, hhd. 

2  hogsheads,   1  pipe, •  •  p. 

2  pipes, 1  tun, T. 

This  table  is  used  for  measuring  wine,  spirits,  cider,  and 
wftter. 


) .  How  many  gills  in  1  quart  ?     In  1  gal.  ?     In  4 
In  6  gals.  ?     In  10  gals.?     In  13  gals.?    In  15  gals.  ? 

2.  How  many  pints  in  1  gal.  ?     In  4  gals.  ?     In  20  gals.  ? 
How  many  qts.  in  1  barrel  ?     In  one  hogshead  ? 

3.  How  many  gallons  in  5  barrels?.    In  8  barrels?     How 
many  gals,  in  half  a  barrel?     In  one  fourth  of  a  barrel  ? 

4.  In  100  gals,  how    many  barrels?      In  300  gals,  how 
many  bis.  ? 

5.  At  14  cents  a  gallon,  what  is  1  qt.  of  vinegar  worth  ?  3 
qts.?  6  qts.?  10  qts.?  15  qts.?  21  qts.?  30  qts.? 

6.  What  is  one  barrel  of  vinegar  worth  at  15  cts.  a  gallon  ? 
How  much,  if  the  price  is  20  cts.  a  gal.  ? 


A  TABLE    OF. ALE   OR  BEER   MEASURE. 

(Used  in  measuring  malt  liquors,  and  milk.) 

2  pints  [pt.]  make     1  quart, qt. 

4  quarts, 1  gallon, gal. 

36  gallons,  •  •  • -1  barrel, bl. 

The  beer  gallon  is  a  little  more  than  one  fifth  larger  than 
the  wine  gallon.  There  are  other  measures  of  beer  besides 
those  in  the  tables  ;  as  the  firkin  of  9  gallons;  the  kilderkin. 


REDUCTION.  39 

18 ;  the  hogshead,  54 ;  but  these  are  not  much  used  in  this 
country.  A  barrel  of  wine  contains  not  quite  three  fourths 
as  much  as  a  barrel  of  beer. 

1.  In  1  bl.  how  many  pints  ?  How  many  pints  in  3  bis.  ? 
How  many  gallons  in  5  bis.  ?  In  12  bis.  ?  In  15  bis.  ?  In 
21  bis.? 

2.  In  100  gallons,  how  many  bis.  ?  How  many  bis.  in  400 
gals.  ?     First  consider  how  many  bis.  there  are  in  360  gals.  ? 

MEASURE  OF  THE  CIRCLE. 

Every  circle  is  supposed  to  have  its  circumference  divided 
into  360  equal  parts,  called  degrees ;  and  each  degree  into  60 
parts,  called  minutes ;  and  each  minute  into  60  parts,  called 
seconds.  Whether  the  circle  is  great  or  small,  it  is  still 
divided  into  360  degrees;  a  degree  therefore  is  always  the 
same  fixed  part  of  the  circumference  of  a  circle,  although  its 
actual  length  is  longer  or  shorter,  according  as  the  circle  is 
great  or  small.  The  line  passing  from  the  centre  to  the  cir- 
cumference is  called  the  radius  of  the  circle.  To  give  you 
some  idea  of  the  length  of  a  degree  in  circles  of  different 
magnitudes,  I  will  state  that,  on  comparing  a  degree  in  any 
circle  with  its  radius,  it  has  been  found  to  be  about  one  fifty- 
eighth  part  of  it.  In  other  words,  58  degrees  on  the  circum- 
ference of  a  circle  are  about  equal  to  the  radius.  If  a  degree 
is  1  inch,  the  radius  of  that  circle  is  58  inches.  If  the  radius 
of  a  carriage  wheel  is  29  inches,  a  degree  on  the  rim  of  the 
same  wheel  wiU  be  half  an  inch. 

If  we  take  for  illustration  one  of  the  largest  sized  water 
wheels,  29  feet  in  diameter,  a  degree  on  its  rim  would  meas- 
ure only  3  inches. 

You  may  enlarge  the  circle  in  your  mind,  tiU  you  suppose 
it  extending  over  a  plain,  with  a  radius  of  58  rods  ;  a  degree 
on  such  a  circle  will  measure  1  rod.  If  the  radius  is  58 
miles,  a  degree  will  measure  1  mile.  Now  the  circle  round 
the  earth  is  so  great  in  extent  that  a  degree  measures  69  J 
miles.  This  may  aid  you  in  forming  a  conception  of  the  vast 
magnitude  of  the  earth. 

Each  of  these  degrees  is  divided  into  60  minutes,  or  geo- 
graphical miles ;  a  geographical  mile  therefore  is  about  one 
sixth  greater  than  a  common  mile.  The  table  of  circular 
measure  is  as  follows : 


40  MENTAL   ARITHMETIC. 

60  seconds   ['']  make 1  minute, '. 

GO  minutes  (or  geog.  miles)    1  degree, *'. 

360  degrees a  circle. 

The  term  miles  instead  of  minutes,  can  be  used  only  in 
reference  to  the  great  circle  of  the  earth. 

As  the  earth  turns  round  on  its  axis  once  in  24  hours, 
every  place  upon  it  passes  in  that  time  through  the  360  de- 
grees of  its  circle ;  and  on  the  equator,  which  is  the  great 
circle,  each  of  these  degrees,  we  have  seen,  is  69 ^^  miles. 

How  swiftly  then  does  a  body  lying  on  the  equator  move 
in  consequence  of  the  daily  .revolution  of  the  earth  ? 

In  24  hours  it  passes  through  360  degrees ;  in  one  hour 
then  it  will  pass  through  one  twenty-fourth  part  as  many, 
which  is  15  degrees.  If  it  pass  through  15  degrees  in  one 
hour,  how  many  minutes  will  it  be  in  passing  through  1  de- 
gree ?  One  fifteenth  of  60  minutes  is  4  minutes.  If  it  pass 
through  a  degree  in  4  minutes,  what  part  of  a  degree  will  it 
pass  through  in  1  minute  ?  -One  fourth  of  a  degree,  or  15 
geographical  miles.  If  it  pa^s  through  15  geographical  miles 
in  1  minute,  in  how  many  seconds  will  it  pass  through  1  geo- 
graphical mile  ?  In  4  seconds ;  and  in  1  second  it  will  pass 
through  one  fourth  of  a  geographical  mile. 

Now  a  geographical  mile  on  the  equator  is,  as  we  have  seen, 
longer  than  a  common  mile.  We  will  here  suppose  it  no 
longer,  but  of  the  same  length,  and  it  appears  that  an  object 
on  the  equator  moves,  as  the  vast  earth  whirls  round  on  its 
axis,  one  quarter  of  a  mile  every  second  of  time.  Reflect 
now,  that,  while  the  surface  of  the  earth  moves  with  such 
amazing  speed,  so  vast  is  its  size,  that  it  occupies  an  entire 
day  and  night  in  turning  once  round. 

If,  as  above  stated,  the  earth  turns  from  west  to  east  at  the 
rate  of  fifteen  degrees  in  an  hour,  we  can,  by  knowing  the 
time  of  day  in  any  place,  ascertain  what  time  it  is  at  a  place 
any  particular  number  of  degrees  east  or  west  of  it.  It  is 
noon  at  any  place  when  the  meridian  of  that  place  passes 
'under  the  sun. 

1.  When  it  is  noon  at  Boston,  what  time  is  it  at  a  place 
15  degrees  west  of  Boston  ?  At  a  place  15  degrees  east  of 
Boston  ? 

2.  When  it  is  12  o'clock  at  Boston,  what  time  is  it  at  a 
place  1  degree  west  of  Boston  ?    At  a  place  1  degree  east  of 


LONGITUDE   AND   TIME.  41 

Boston  ?  At  a  place  2  degrees  west  of  Boston  ?  At  a  place 
2  degrees  east  of  Boston  ?  3  degrees  east  ?  3  degrees  west  ? 
4  degrees  east?  4 degrees  west?  5  degrees  east?  5  degrees 
west? 

3.  Indianapolis  is  15  degrees  west  of  Boston;  when  it  is 
noon  at  Boston,  what  time  is  it  at  Indianapolis  ?  When  it  is 
sunset  at  Boston,  where  will  the  sun  be  at  Indianapolis  ? 

4.  Niagara  Falls  is  8  degrees  west  of  Boston ;  when  it  is 
noon  at  Boston,  what  time  is  it  at  Niagara  Falls  ?  When  it 
is  4  o'clock  at  Niagara  Falls,  what  time  is  it  at  Boston  ? 

5.  Washington  city  is  6  degrees  west  of  Boston  ;  if  you  set 
your  watch  with  the  sun  at  Boston,  and  then  carry  it  to 
Washington,  your  watch  keeping  accurate  time  all  the  while, 
when  you  arrive  at  Washington,  will  it  be  too  fast  or  too 
slow  ?  and  how  much  ? 

6.  Two  travelers  met  at  a  public  house ;  when  one  of  them 
said  to  the  other,  "  Friend,  are  you  traveling  east  or  west  ?  " 
"  I  am  direct  from  home,"  said  the  other,  "  where  my  watch 
agrees  exactly  with  the  sun,  but  here  I  find  it  is  10  minutes 
too  fast :  now  if  you  can  tell  which  way  I  am  traveling  you  are 
welcome  to  know." 

Had  he  traveled  east,  or  west  ?   and  how  far  ? 

7.  Boston  is  71  degrees  west  of  London;  when  it  is  noon 
at  Boston,  what  time  is  it  in  London  ? 

8.  The  English  convicts  are  transported  to  Botany  Bay, 
150  degrees  east  of  London ;  when  it  is  noon  at  London,  what 
time  is  it  in  Botany  Bay  ? 

9.  Enghsh  traders  are  settled  on  Columbia  river,  120  de- 
grees west  from  London ;  what  time  is  it  there  when  it  is 
noon  in  London  ? 

10.  If  a  man  is  on  the  equator,  which  way  must  he  travel, 
and  how  many  geographical  miles,  to  have  the  day  4  minutes 
longer  than  24  hours?  How  far  to  have  the  day  2  minutes 
longer  ?  How  far  to  have  it  one  minute  longer  ?  How  far 
must  he  travel  to  have  the  day  one  minute  and  a  half  longer  ? 
Which  way  must  he  travel  and  how  far,  to  have  the  day  one 
minute  shorter  ?  2  minutes  shorter  ?  5  minutes  shorter  ? 

11.  Suppose  two  birds  start  from  the  same  place  on  the 
equator,  and  fly,  one  east  and  the  other  west,  at  the  rate  of 
60  geographical  miles  an  hour,  and  at  the  end  of  the  hour  it 
is  just  sunset  to  the  bird  flying  east;  how  high  is  the  sun 
then  at  the  place  where  the  other  bird  is  ? 

4* 


42  MENTAL   ARITHMETIC. 

How  high  was  the  sun  at  the  place  of  their  starting,  when 
they  set  out  ? 

12.  A  shipmaster  sails  from  New  York  for  Europe,  and 
for  three  days  it  is  so  cloudy  that  he  cannot  see  the  sun ;  on 
the  fourth  day  he  takes  an  observation  of  the  sun  at  noon ; 
and  by  his  chronometer,  which  gives  the  New  York  time,  it 
is  half  past  eleven ;  how  many  degrees  east  from  New  York 
has  he  sailed  ? 

In  what  longitude  is  he  then,  if  New  York  is  74°  1'  west 
from  Greenwich? 


SECTION    VII. 

PRIME  NUMBERS. 

Numbers  may  be  divided  into  two  great  classes.  The  first 
class  comprises  such  numbers  as  cannot  be  formed  by  the  mul- 
tiplication of  any  two  or  more  numbers  together,  as  1,  2,  3,  5, 
11,  17.  These  are  called  Prime  numbers.  The  other  class 
may  be  formed  by  multiplying  two  or  more  numbers  together, 
as  4,  which  is  formed  by  multiplying  2  by  2 ;  6,  which  is 
equal  to  2X3  ;  10,  which  is  equal  to  2X5,  &c.  These  are 
called  Composite  numbers.  These  may  always  be  formed  by 
multiplying  two  or  more  prime  numbers  together.  Thus  all 
numbers  are  either  Prime,  or  are  formed  by  the  multiplica- 
tion of  Prime  numbers  together. 

In  separating  numbers  into  their  factors  care  should  be 
taken  that  the  factors  be  all  prime.  Thus  in  resolving  30 
into  its  factors  we  may  say  it  is  formed  by  multiplying  5  by 
6,  but  this  is  not  sufficient,  for  6  is  not  prime  ;  it  is  formed  of 
the  factors  2  and  3.  The  prime  factors  of  30  therefore  are 
2,  3  and  5.  We  may  say  that  30  is  formed  of  the  factors  3 
and  10,  but  here  again  the  analysis  is  not  complete,  for  10 
is  not  prime ;  it  is  composed  of  the  factors  2  and  5.  Thus 
we  are  brought  to  the  same  three  factors  as  before,  namely, 
2,  3  and  5. 

The  following  table  of  numbers  from  1  to  100  will  show 
what  of  them  are  prime,  and  what  are  the  prime  factors  of 
those  which  are  composite.    This  table  should  be  carefully 


PRIME   AND    COMPOSITE    NUMBERS. 


43 


studied  and  made  perfectly  familiar.  The  analysis  of  compo- 
site numbers  into  their  prime  factors  lies  at  the  foundation  of 
so^e  of  the  most  important  operations  in  numbers,  and  affords 
ai»  insight  into  some  of  the  most  intricate  rules  of  Arithmetic. 


1  prime. 

2  prime. 

3  prime. 
4=2X2. 
5  prime. 
6=3X2. 
7  prime. 
8=2X2X2. 
9=3X3. 

10=2X5. 

11  prime. 

12=2X2X3. 

13  prime. 

14=2X7. 

15=3X5. 

16=2X2X2X2. 

17  prime. 

18=2X3X3. 

19  prime. 

20=2X2X5. 

21=3X7. 

22=2X11. 

23  prime. 

24=2X2X2X3. 

25=5X5. 

26=2X13. 

27=3X3X3. 

28=2X2X7. 

29  prime. 

30=2X3X5. 

31  prime. 

32=2X2X2X2X2. 

33=3X11. 

34=2X17. 

35=5X7. 

36=2X2X3X3. 

37  prime. 


38=2X19. 

39=3X13. 

40=2X2X2X5. 

41  prime. 

42=2X3X7. 

43  prime. 

44=2X2X11. 

45=5X3X3. 

46=2X23. 

47  prime. 

48=2X2X2X2X3. 

49=7X7. 

50=2X5X5. 

5l=r3Xl7. 

52=2X2X13. 

53  prime. 

54=2X3X3X3. 

55=5X11. 

56=2X2X2X7. 

57=3X19. 

58=2X29. 

59  prime. 

60=2X2X3X5. 

61  prime. 

62=2X31. 

63=3X3X7. 

64=2X2X2X2X2X2. 

65=5X13. 

66=2X3X11. 

67  prime. 

68=2X2X17. 

69=3X23. 

70=2X5X7. 

71  ^rime. 

72=2X2X2X3X3. 

73  prime. 

74=2X37. 


44 


-      MENTAL   ARITHMETIC. 


75=3x5X5. 

76=.2X2X19. 

77=7X11. 

78=2X3X13. 

79  prime. 

80=2X2X2X2X5. 

81z:r3X3X3X3. 

82=2X41. 

83  prime. 

84=2X2X3X7. 

85=5X17. 

86=2X43. 

€7=3X29. 


88=2X2X2X11. 
89  prime. 
90=2X3X3X5. 
91=7X13. 
92:1=2X2X23. 
93=3X31. 
94=2X47. 
95=5X19. 

96=2X2X2X2X2X3. 
97  prime. 
98=2X7X7. 
99=3X3X11. 
100=2X2X5X5. 


On  examining  this  table  several  things  may  be  observed. 

1.  All  the  even  numbers  are  composite ;  for  they  are  all 
divisible  by  2.  So  it  appears  in  the  table,  with  the  exception 
of  the  number  2,  which  is  regarded  as  prime  because  it  is 
divisible  only  by  itself. 

2.  Several  of  the  numbers  given  above  are  powers  of  their 
prime  factors.  Thus  4  is  the  2d  power  of  2,  8  the  3d  power 
of  2,  16  the  4th  power  of  2,  32  the  5th,  64  the  6th.  9,  27 
and  81,  are  the  2d,  3d  and  4th  powers  of  3.  25  is  the  2d 
power  of  5,  49  the  2d  power  of  7. 

3.  If  you  double  the  number  of  times  a  factor  is  taken, 
you  obtain  the  square  of  the  number  they  at  first  made. 
Thus  4  is  obtained  by  taking  2  twice  as  a  factor.  If  you  take 
it  twice  as  many  times,  that  is,  4  times  as  a  factor,  you  obtain 
16,  which  is  the  square  of  4. 

9  is  obtained  by  taking  3  twice  as  a  factor.  If  you  double 
the  number  of  times  it  is  taken,  thus,  3X3X3X3,  you  ob- 
tain the  square  of  9. 

8  is  obtained  by  taking  2  three  times  as  a  factor.  If  you 
take  it  6  times  you  obtain  64,  the  square  of  8. 

So  universally,  if  you  double  the  number  of  times  a  factor 
is  taken  to  produce  a  certain  number,  you  obtain,  not  twice 
that  number,  but  the  square  of  it. 

I  will  make  a  single  remark  here  about  the  prime  numbers, 
and  then  call  your  attention  to  the  composite  numbers. 

Since  the  'prime  numbers  are  not  formed  by  multiplying 
any  two  or  more  numbers  together,  they  cannot  be  divided  by 
any  number.     You  will  observe,  however,  that  any  number 


PROPERTIES    OF   PRIME   AND    COMPOSITE   NUMBERS.      45 

whatever  may  be  divided  by  itself,  and  may  also  be  divided 
by  1 ;  but  1  is  a  unit  and  not  a  number;  and  by  dividing  a 
number  by  itself,  or  by  1,  you  obtain  no  new  number.  Di- 
viding the  number  by  itself  you  obtain  1,  and  dividing  by  1, 
you  obtain  the  number  itself.  Such  an  operation,  therefore, 
brings  out  nothing  new.  It  is  only  another  way  of  express- 
ing what  was  just  as  plain  before.  In  the  same  way  we  may 
sometimes  regard  a  number  as  produced  by  multiplying  itself 
into  1 ;  thus,  7=7  X 1 ;  but  this  is  not  multiplication,  but 
only  an  expression  in  the  form  of  multiplication.  It  pro- 
duces no  new  number,  and  is  employed  only  for  convenience 
in  order  to  make  the  reasoning  more  plain. 

Composite  numbers  can  be  divided  by  their  factors.  Thus 
you  can  divide  10  either  by  2  or  by  5,  and  by  no  other  num- 
ber. If  you  divide  by  2,  you  obtain  5  for  the  answer,  or 
quotient;  if  you  divide  by  5  you  obtain  2  for  the  answer. 
Dividing  by  a  number  then,  is  the  same  as  erasing  that  num- 
ber as  a  factor,  and  will  always  give  for  the  answer  the  other 
factor,  or  factors.  Thus  dividing  10  by  2  you  may  represent 
thus,  ^X5,  leaving  the  factor  5  for  the  answer:  dividing  10 
by  5,  thus,  2X^j  leaving  2  for  the  answer.  Divide  21  by  3, 
thus,  ^X7.     Divide  1 2  by  3  thus,  4X3,  or^X^X3. 

It  is  plain,  therefore,  that  if  you  express  any  number  by  its 
factors  you  can  at  once  see  what  numbers  you  can  divide  it 
by.  You  can  divide  it  by  each  of  its  prime  factors,  or  by  any 
combination  of  them,  and  by  no  other  number.  Thus  6= 
2X3,  you  can  divide  by  2  or  by  3  :  8=2X2X2,  you  can 
divide  by  2,  and  that  quotient  by  2,  and  that  by  2  again ; 
30=2X3X5,  you  can  divide  by  2,  or  3,  or  5,  or  by  any  two 
of  them  combined. 

Any  composite  numbers  may  be  divided  by  any  of  its  prime 
factors,  or  by  any  combination  of  them. 

By  what  numbers  can  you  divide  15?  18?  20?  21?  26? 
27?  36?  42?  46?  48?  49?  50? 

Sometimes  we  have  two  numbers,  and  we  wish  to  know  if 
there  is  any  number  that  will  divide  them  both.  This  we 
can  ascertain  if  we  express  each  number  by  means  of  its 
prime  factors,  and  then  see  if  the  same  factor  is  found  in  both : 
if  vSO,  they  are  both  divisible  by  that  number.  Thus,  if  we 
wish  to  know  whether  any  number  will  divide  both  9  and  15, 
we  express  them  thus,  3X3  and  3X5.  Now  3  appears  as  a 
factor  in  both ;    they  can  both   therefore  be  divided  by  3. 


46  MENTAL    ARITHMETIC. 

This  number  3  is  called  the  common  divisor,  because  it  is  a 
divisor  common  to  several  numbers.  If  we  wish  to  know 
whether .  any  number  will  divide  both  15  and  8,  we  express 
15  bj  its  factors,  5x3 ;  and  8  by  its  factors,  2X2X2.  Now 
there  is  no  factor  common  to  both ;  no  number  therefore  will 
divide  them  both ;  in  other  words  they  have  no  common 
divisor.  Numbers  which  have  no  common  divisor  are  said 
to  be  prime  to  each  other.  They  may  be  composite  consid-' 
ered  by  themselves,  as  is  the  case  with  8  and  15,  but  if  they 
have  no  common  divisor  they  are  said  to  be  prime  to  each 
other.  Numbers  which  have  a  common  divisor  are  said  to  be 
composite  to  each  other.  If  there  are  more  than  two  numbers 
they  must  be  treated  in  the  same  way.  Each  must  be  writ- 
ten in  the  form  of  its  prime  factors,  and  then,  if  any  one 
number  appears  as  a  factor  in  them  all,  they  are  divisible  by 
that. 

Is  there  any  common  divisor  to  9,  14  and  27?  Written 
in  the  form  of  their  factors  they  stand  thus,  3X3.  2X7. 
3X3X3.  They  have  therefore  no  common  divisor ;  for, 
though  3  or  9  will  divide  both  the  first  and  the  third  number, 
it  will  not  divide  the  second  ;  and  neither  2  nor  7,  which  are 
the  factors  of  the  second  number,  appear  in  the  first  or  third. 
9,  14  and  27  are,  therefore,  prime  to  each  other. 

What  is  the  common  divisor  of  15  and  27  ?  of  14  and  22  ? 
of  21  and  49  ?  of  35  and  28  ?  of  6  and   21  ? 

Let  us  now  take  the  following  question:  What  is  the 
common  divisor  of  18  and  30  ?  By  inspecting  their  factors 
2X3X3,  and  2X3X5,  we  find  that  2X3  or  6,  is  common  to 
both ;  6  is  therefore  the  greatest  common  divisor. 

What  is  the  greatest  common  divisor  of  18  and  27?  of  4, 
8  and  36?  of  15  and  45?  of  27  and  45?  of  40,  64,  and  16? 
of  44  and  24  ?  of  75  and  15  ?  of  80  and  100  ?  of  60  and  24  ? 
of  35,  21  and  49  ?  of  15  and  50? 

We  have  seen  that  a  composite  number  can  be  divided 
only  by  its  factors  ;  and  that  prime  numbers  cannot  be  divided 
at  all.  It  is  frequently  necessary,  however,  to  attempt  the 
division  of  prime  numbers  ;  and  to  divide  composite  numbers 
by  some  number  different  from  their  factors.  For  example, 
we  may  wish  to  divide  9  by  4,  or  to  obtain  one  fourth  of  9. 
Now  4  is  not  a  factor  of  9,  and  the  actual  division  of  9  by  4 
is,  strictly  speaking,  impossible.  We  proceed  in  this  way. 
We  divide  8  by  4,  and  obtain  2  for  the  answer,  and  we  have 


INCOMPLETE   DIVISION.      TRACTIONS.  47 

a  remainder  of  1  which  we  have  not  divided.  To  show  that 
w^e  design  this  to  be  divided  bj  4  we  write  the  4  "under  it 
with  a  Une  between,  thus,  ^.  ~  In  this  way  we  indicate  plainly 
enough  what  the  answer  is,  although  we  have  no  one  figure 
that  will  express  it. 

This  operation  introduces  us  to  a  new  class  of  quantities 
called  Fractions,  fractions  are  expressions  for  quantities  less 
than  a  unit.  The  word  Fractions  here  means  the  same  as 
broken  numbers.  In  this  class  of  expressions  each  unit  is 
regarded  as  broken  up,  or  divided  into  a  number  of  parts. 
The  figure  below  the  line  shows  into  how  many  parts  the 
unit  is  divided ;  the  figure  above  the  line  shows  how  many 
of  those  parts  are  taken ;  (or,  what  is  just  equivalent,  the 
number  above  the  line  is  regarded  as  divided  by  the  number 
below  it.)  The  fraction  y  indicates  that  each  of  the  3  units  is 
regarded  as  divided  into  7  equal  parts ;  and  that  one  of  these 
parts  is  taken  from  each  of  them.  The  number  below  the 
line  is  called  the  Denominator ;  that  above  the  line,  the  Nu- 
merator. If  the  Numerator  is  just  equal  to  the  Denomina- 
tor, as  I,  J,  7,  the  value  of  the  fraction  is  just  equal  to  1.  If 
the  Numerator  is  smaller  than  the  Denominator,  the  value  of  - 
the  fraction  is  less  than  1,  and  is  called  a  proper  fraction  ;  if 
the  Numerator  is  greater  than  the  Denominator,  the  value  of 
the  fraction  is  greater  than  1,  and  is  called  an  improper 
fraction.  This,  however,  may  always  be  changed  to  a  whole 
number,  or  a  whole  number  and  a  proper  fraction.  Hence 
the  propriety  of  the  definition,  that  fractions  are  expressions 
for  quantities  less  than  unity. 

Questions* 

What  is  meant  by  a  Common  Divisor  ? 

What  is  meant  by  the  greatest  Common  Divisor  ? 

When  are  numbers  prime  to  each  other  ? 

When  are  numbers  composite  to  each  other  ? 

What  is  the  process  of  dividing  13  by  4  ? 

In  dividing  16  by  5  ?     In  dividing  25  by  6  ? 

What  are  Fractions'^ 

Explain  what  is  signified  by  each  of  the  numbers  in  the 
fraction  |.   In  f    In  A-    In  y\.    In  |.    In  ||. 


48  MENTAL   ARITHMETIC. 

A  man  bought  a  barrel  of  flour,  and  gave  away  two  fifths 
of  it ;  what  fraction  will  express  what  he  gave  away  ?  What 
fraction  will  express  what  he  kept  ? 

A  man  bought  a  load  of  hay,  and  sold  two  elevenths  of  it ; 
what  fraction  will  express  what  he  sold  ?  What  fraction  wiU 
express  what  he  kept  ? 

What  is  a  proper  fraction  ?     Give  an  example. 

What  is  an  improper  fraction  ?     Give  an  example. 

When  is  the  value  of  a  fraction  just  equal  to  1  ? 


SECTION    VIII. 

MULTIPLICATION  AND  DIVISION  OF  FRACTIONS. 

We  have  seen  that  a  fraction  is  not  a  simple  expression,  but 
composed  of  two  numbers ;  and  its  value  cannot  be  determined 
by  one  of  these  numbers  alone,  but  by  both  taken  in  connec- 
tion. By  looking  at  the  numerator,  you  cannot  tell  the  value 
of  the  fraction  unless  you  know  what  the  denominator  is. 
By  looking  at  the  denominator  you  cannot  tell  the  value  of 
the  fraction,  unless  you  know  what  the  numerator  is. 

Let  us  now  observe  the  effect  of  altering  one  of  the  terms 
of  the  fraction  without  altering  the  other.  We  will  take  the 
fraction  f .  If  we  increase  the  numerator  by  1,  making  it  f , 
we  increase  the  value  of  the  fraction,  for  we  take  one  fifth 
more  than  we  had  before.  So,  if  we  multiply  the  numerator 
by  2,  making  it  J,  we  double  the  value  of  the  fraction  ;  and  so 
of  any  other  numbers,  if  we  multiply  the  numerator,  we  mul- 
tiply the  value  of  the  fraction.  And,  by  the  same  reasoning, 
if  we  divide  the  numerator  by  2,  we  divide  the  fraction  by  2, 
for  J  is  plainly  one  half  as  great  as  J.  So  of  all  other  num- 
bers, by  dividing  the  numerator  we  divide  the  fraction. 

Let  us  now  observe  the  effect  of  altering  the  denominator. 
If  we  increase  the  denominator  of  the  fraction  f  by  1,  making 
it  f ,  we  have  not  increased  the  fraction,  but  diminished  it,  for 
one  sixth  is  less  than  one  fifth,  and  any  number  of  sixths  are 
less  than  the  same  number  of  fifths.  We  will  multiply  the 
denominator  ,of  the  fraction  §  by  2,  making  f*^.  Wliat  effect 
has  been  produced  on  the  value  of  the  fraction  ?     One  tenth 


FRACTIONS.  49 

is  half  as  great  as  one  fifth ;  and  two  tenths  are  half  as  great  as 
two  fifths.  The  fraction  is  therefore  half  as  great  as  it  was 
before  ;  that  is,  it  has  been  divided  bj  2.  Multiplying  the 
denominator,  therefore,  divides  the  value  of  the  fraction. 

We  will  now  divide  the  denominator.  Take  the  fraction  | ; 
dividing  the  denominator  by  2,  we  have  |.  Now  as  this  is 
twice  as  great  as  |,  we  have  multiplied  the  fraction,  by  di- 
viding the  denominator. 

There  are,  then,  two  ways  of  multiplying  a  fraction. 
We  may  multiply  the  numerator ;  or,  if  the  multiplier  is  a 
factor  of  the  denominator,  we  may  divide  the  denominator. 
Thus,  to  multiply  |  by  2,  we  may  multiply  the  numerator, 
which  gives  f ,  or  divide  the  denominator,  which  gives  |,  equal 
to|. 

To  divide  a  fraction,  we  may  either  divide  the  numerator,  if 
the  divisor  is  a  factor  of  it ;  or  we  may  multiply  the  denomi- 
nator. Thus,  to  divide  f  by  3,  we  may  divide  the  numerator, 
giving  f ,  or  we  may  multiply  the  denominator,  which  gives 
^®y,  which  is  equal  to  f . 

We  will  now  multiply  both  terms  of  the  fraction  by  the 
same  number.  Multiplying  both  terms  of  the  fraction  f  by  3, 
we  have  |.  Here  the  denominator,  expressing  the  number 
of  parts  into  which  the  unit  is  divided,  is  three  times  as  great 
as  it  was  before,  consequently  each  of  the  parts  is  only  one 
third  as  great ;  but  the  numerator  has  also  been  multiplied  by 
three,  so  that  three  times  as  many  parts  are  taken,  and  this 
makes  the  value  of  the  fraction  just  equal  to  what  it  was 
before.  So  we  may  multiply  by  any  number  whatever,  both 
terms  of  the  fraction  §,  and  the  value  will  still  be  the  same  as 
before ;  for  example,  |,  f ,  y\,  yf ,  yf,  each  of  which  is  equal 
to  §.  We  may  then  at  any  time  multiply  both  terms  of  a 
fraction  by  the  same  number,  without  altering  the  value  of 
the  fraction.  By  the  same  reasoning  we  may  divide  both 
terms  of  a  fraction  by  the  same  number  without  altering  its 
value.  Taking  the  examples  above,  we  may  divide  the  terms 
of  f  by  2,  and  we  obtain  f ;  dividing  the  terms  of  |  by  3  gives 
us  §,  and  so  of  the  others ;  f  is  the  same  fraction  as  |,  f ,  y^, 
&c.,  but  it  is  expressed  in  lower  terms,  and  therefore  is  more 
convenient.  It  is  easier  to  write  J  than  it  is  to  write  Jf , 
though  both  have  the  same  value. 

To  reduce  a  fraction  to  its  lowest  terms,  we  divide  both  the 
numerator  and  denominator  by  their  greatest  common  divisor. 
5 


50  MENTAL    ARITHMETIC. 

To  find  the  greatest  common  divisor,  separate  each  term  into 
its  prime  factors,  and  erase  those  which  are  common  to  both. 
The  remaining  factors  will  express  the  value  of  the  fraction 
in  its  lowest  terms. 

Treating  the  above  fractions  in  this  way  thej  appear  thus, 

4_^X2  6_2X^    8_^X^X2  10_2X^  12_^X2Xg( 

6~^X3'  9"~3X3'  12~^X^X3'  l5~3X?'  18""^X3X$' 

leaving  in  each  case  |-. 

In  how  many  ways  can  you  obtain  the  answer  to  the  follow- 
ing questions  ?  JX2?  1X3?  1X4?  t\X2? 

In  how  many  ways  can  you  obtain  the  answer  to  the  follow- 
ing?  fX3?  1x2?  1x4?  t\X6?  AX5?  t\X3? 

In  how  many  ways  can  you  obtain  the  answer  to  the  follow- 
ing? f^3?  f-r-4?  /^-~.3?  1*^2?  ifH-4? 

In  how  many  ways  can  you  obtain  an  answer  to  the  foUow- 

Reduce  to  their  lowest  terms  each  of  the  following'  fractions, 

e     14      3       20     40     16     18     24     26       6        9       16       8       30     70 
H>  ITTJ  S^J  ^^?  ¥0J  ST^)  TSi  57>  SS^  TF>  T3")  ^^i  '2'^y  ?F5  ^Z- 


/    TO  FIND  THE  DIVISORS  OF  NUMBERS. 

Reduce  the  fraction  -^J J  to  its  lowest  terms  ? 

You  will  not  see  immediately  that  these  two  numbers  have 
any  common  divisor.  To  assist  you  to  reduce  fractions  of  this 
kind,  something  will  here  be  said  about  the  way  of  finding  the 
divisors  of  numbers.  Let  us  first  inquire  what  numbers  can 
be  divided  by  2. 

We  have  seen  that  all  even  numbers,  and  only  those,  can 
be  divided  by  2. 

What  numbers  can  be  divided  by  4  ? 

If  you  examine  you  will  find  that  all  even  tens  are  divisible 
by  4,  as  20,  40,  60,  &c.  K,  therefore,  the  tens  are  even,  and 
the  units  are  divisible  by  4,  then  the  whole  is  divisible  by  4. 
But  the  only  unit  numbers  divisible  by  4  are  4  and  8  ;  there- 
fore if  the  tens  are  even,  and  the  unit  number  is  4  or  8,  the 
whole  is  divisible  by  4 ;  as  84,  88  ;  124,  128  ;  148,  364,  &c. 

Again,  as  10  when  divided  by  4  leaves  a  remainder  of  2, 
any  odd  number  of  tens  will  do  the  same,  as  30,  50,  70,  90 ; 
for  every  odd  number  of  tens  is  an  even  number  of  tens-(-10. 
If,  then,  the  number  of  tens  is  odd,  the  units  must  be  two  less 


FRACTIONS.  51 

than  4  or  8,  in  order  to  be  divisible  bj  4.  That  is,  if  the 
tens  are  odd,  and  the  units  2  or  6,  the  whole  is  divisible  by  4 ; 
as  72,  96,  52,  &c. 

Are  the  following  even  numbers  divisible  by  4  or  only  by  2 ; 
and  why  ?     126,  82,  94,  92,  138,  156,  346,  548,  76,  58,  392. 

What  numbers  can  be  divided  by  8  ? 

As  100  divided  by  8  leaves  a  remainder  of  4  (8X12=96,) 
it  follows  that  200  will  be  exactly  divisible  by  8,  for  the  two 
remainders  of  4  will  make  8.  If  200  is  divisible  by  8,  it 
follows  that  all  even  hundreds  are  divisible  by  8  ;  as  400, 
600,  1400,  &c. 

If,  therefore,  the  hundreds  are  even  and  the  tens  and  units 
are  divisible  by  8,  the  whole  number  will  be  'divisible  by  8 ; 
ii<  248,  672, 1456,  &c. 

Again,  if  the  hundreds  are  odd  and  the  tens  and  units  are 
4  less  than- some  multiple  of  8,  the  whole  number  will  be  di- 
visible by  8 ;  for  the  odd  hundred,  divided  by  8,  leaves  a  re- 
maiiuler  of  4;  and  this,  added  to  the  tens  and  units,  will  make 
an  exact  multiple  of  8. 

Are  the  following  numbers  divisible  by  8,  or  by  4 ;  and 
why?  444,  944,  136,  1328,  712,  532,  816,  516,  384,  128, 
1236. 

What  numbers  are  divisible  by  5  ?  All  tens  are  divisible 
by  5  ;  consequently  if  the  unit  figure  is  5  or  0,  the  whole 
number  is  divisible  by  5. 

What  numbers  are  divisible  by  3  ?  By  examining  the 
multiples  of  3  we  shall  find  this  singular  fact,  that  the  sum 
of -the  figures  which  express  any  multiple  of  3  is  itself  a  mul- 
tiple of  3.  Take  the  multiples  of  three  from  12  to  24  ;  12, 
15,  18,  21,  24  ;  by  adding  the  figures  which  express  any  one 
of  these  multiples  we  find  that  the  sum  is  a  multiple  of  3. 
The  figures  of  12  added  are  1+2=3,  of  15  are  1+5=6,  of 
18  are  1+8=9,  of  21  are  2+1=3,  of  24  are  2+4=6.  The 
same  is  true  of  all  multiples  of  3. 

It  will  also  be  found  that  if  you  add  the  figures  of  any 
number  and  the  sum  is  a  multiple  of  three,  the  whole  number 
is  a  multiple  of  three.  To  know,  then,  if  a  number  is  a 
multiple  of  3,  add  together  the  figures  that  express  the  num- 
ber, and  if  the  sum  is  a  multiple  of  3,  the  whole  number  is  a 
multiple  of  3. 

Are  the  following  numbers  divisible  by  3  ?  471, 59, 115,  642, 
624,  138,  234,  742,  894. 


52  MENTAL    ARITHMETIC. 

It  follows  from  what  has  been  said,  that  if  any  number  is 
divisible  by  3,  any  other  number  expressed  by  the  same  fig- 
ures differently  arranged  will  also  be  divisible  by  3  ;  for  the 
sum  made  by  adding  the  figures  will  be  the  same  in  whatever 
order  they  are  taken. 

Thus,  if  936  is  divisible  by  3  ;  369,  396,  963,  639,  693  are 
each  divisible  by  3. 

We  will  next  inquire  what  numbers  are  divisible  by  6. 
As  6=2X3,  any  number  that  is  divisible  by  2  and  by  3  is 
divisible  by  6.  You  have  learned  what  numbers  are  divisible 
by  3,  and  what  by  2.  If  a  number  combines  both  these  con- 
ditions, it  is  divisible  by  6  ;  that  is,  all  numbers  are  divisible 
by  6,  the  sum  of  whose  figures  is  a  multiple  of  3,  and  whose 
last  figure  is  an  even  number. 

What  combinations  of  the  figures  1,  2,  3,  will  give  numbers 
divisible  by  6  ;  and  what  by  3  only  ? 

Next  let  us  inquire  what  numbers  are  divisible  by  9. 

If  the  figures  which  express  any  multiple  of  9,  as  18,  27, 
36,  45,  54,  be  added  together,  the  sum  will  be  a  multiple  of  9. 

Also,  if  the  figures  of  any  number  be  added  together,  and 
the  sum  is  a  multiple  of  9,  the  whole  number  is  divisible  by  9. 

Are  the  following  numbers  divisible  by  9  ?  and  why  ?  936, 
972,  396,  423,387,  527,  441,  416,  315,  756. 

Any  number  divisible  by  9  and  by  2  is  divisible  by  9X2, 
or  18  ;  M'hich  of  the  above  numbers  are  divisible  by  18  ? 

Any  number  divisible  by  9  and  by  4  is  divisible  by  9X4, 
or  36  ;  which  of  the  above  numbers  are  divisible  by  36  ? 

Any  number  divisible  by  9  and  by  8  is  divisible  by  9X8, 
or  72  ;  is  either  of  the  above  numbers  divisible  by  72  ? 

Any  number  divisible  by  9  and  by  5  is  divisible  by  9X5, 
or  45  ;  which  of  the  above  numbers  is  divisi])ie  by  45  ? 

Whatare  divisors  of  124?  of  176  ?  of  252  ?  of  384?  of 
153  ?   of  186  ?   of  207  ?   of  702  ?   of  4041  ? 

We  will  now  return  to  the  fraction  that  was  first  given. 
Reduce  yJJ  to  its  lowest  terms. 

Reduce  to  lowest  terms,  ||^ ;   f  || ;   |§f . 

Reduce  to  lowest  terms,  f||;   %\%',   f||. 


FRACTIONS.  53 

SECTION    IX. 

MULTIPLICATJON  OF  FRACTIONS  BY  FE ACTIONS. 

We  have  seen  how  we  may  multiply  or  divide  a  fraction 
by  a  whole  number.  We  will  now  inquire  how  we  can  multi- 
ply or  divide  one  fraction  by  another.  Let  us  multiply  f  by  f . 
First  multiply  f  by  2,  which  gives  y  for  the  answer.  But 
here  we  have  multiplied  by  2,  instead  of  the  real  multiplier,  §. 
Now  2  is  5  times  greater  than  § ;  the  product  ^  then  is  5  times 
greater  than  it  should  be.  It  must  therefore  be  divided  by  5. 
We  divide  f  by  5  by  multiplying  the  denominator  by  5,  giving 
^3"  for  the  answer. 

In  the  same  way  multiply  f  by  yy.  yXJ-  f  Xj. 

DIVISION  OF  FRACTIONS  BY  FRACTIONS. 

Let  us  now  divide  f  by  f.  First  divide  f  by  3.  This  we 
do  by  multiplying  the  denominator  by  3,  giving  for  the  an- 
swer j*3-.  Here,  however,  we  have  divided  by  3,  instead  of  the 
true  divisor,  f .  We  have  used  a  divisor  seven  times  too  large. 
The  quotient,  therefore,  will  be  seven  times  too  small ;  y%-  must 
therefore  be  multiplied  by  7,  making  the  answer  ff .  In  the 
same  way  perform  the  following :    f-i-f  •    f -i-|.    f -t-|.    j  ',  7. 

The  above  analysis  shows  the  grounds  of  the  rules  usually 
given  in  Arithmetics  for  the  multiplication  and  division  of 
fractions. 

For  Multiplication,  multiply  the  numerators  together  for  a 
new  numerator,  and  the  denominators  for  a  new  denominator. 

For  Division,  invert  the  divisor  and  proceed  as  in  multipli- 
cation. 

Sometimes  we  wish  to  find  the  value  of  a  compound  frac- 
tion, as  §  of  f ;  in  such  cases  we  may  understand  the  sign  of 
multiplication,  X  ?  to  stand  in  the  place  of  the  word  of,  and 
treat  it  as  a  case  of  multiplication.  For  in  the  above  example 
it  is  plain  that  one  third  of  |  is  j\,  and  two  thirds  is  twice  as 
much,  that  is,  y^. 

Wliat  is  I  of  I  of  f  ?  Multiplying  as  we  have  done  above, 
we  have  for  the  answer  y^.  But  {his  operation  may  be 
shortened.  We  see  that  4  appears  as  a  factor  both  in  the 
5* 


54  MENTAL   ARITHMETIC. 

numerator  and  the  denominator.  We  may  then  cancel  them 
both,  which  will  have  the  same  effect  as  dividing  both  terms 
of  the  answer  by  4.  Again,  3  appears  in  both  the  numerator 
and  the  denominator,  for  in  the  denominator  it  is  a  factor  of  9. 
We  may  therefore  cancel  3  in  both  terms. 

The  question  will  then  appear  thus,  qX-vX^,  substituting 

3  in  place  of  the  9.  Multiplying  together  the  terms  that  now 
remain,  we  have  f  for  tl;e  answer.  This  is  the  same  fraction 
as  yW.  If  you  separate  the  terms  of  y/^  into  their  prime 
factors,  and  cancel  what  are.  common  to  both,  the  remaining 
factors  will  give  the  fraction  |. 

Multiply  the  fractions  IXIXtIXt)  writing  the  terms 
that  are  composite  in  the  form  of  their  prime  factors,  and 
canceling  factors  that  are  common  in  both,  it  will  stand 

^X^X2><?^X3X?'<r  which  gives  ,V 
Multiply  IXIXf.    liXllXf. 
Multiply  |?XtV   liX-,V    J|X|X|. 

TO  MULTIPLY  OE  DIVIDE  WHOLE  NUMBERS  BY  FEACTIONS. 

The  above  examples  will  show  how  to  multiply  or  divide  a 
whole  number  by  a  fraction. 

Multiply  7  by  |.  Multiplying  7  by  4  gives  28,  which  is  5 
times  too  great,  because  4  is  five  times  greater  than  J.  We 
must  therefore  divide  the  answer  by  5,  thus  \®.  As  this  is 
more  than  1,  we  can  reduce  it  to  a  whole  number  and  a  frac- 
tion. As  |-  is  equal  to  1,  ^/  will  be  equal  to  5  ;  ^-^  therefore 
is  equal  to  5|. 

In  this  way  multiply  6  by  J.    9  by  |.    8  by  y. 

This  operation  is  in  fact  the  same  as  multiplying  a  fraction 
by  a  whole  number,  which  has  been  treated  of  already. 

Let  us  next  divide  7  by  |.  Dividing  7  by  3  we  have  J  ; 
here,  however,  we  have  divided  by  a  number  4  times  too  great, 
for  3  is  four  times  greater  than  |,  If  the  divisor  is  4  times 
too  great,  the  quotient  will  be  4  times  too  small ;  J,  therefore, 
must  be  multiplied  by  4,  giving  ^-g  for  the  answer. 

Divide  8  by  4.    9-^|.    11-^.|.    10~fV 

To  reduce  an  improper  fraction,  as  ^:j  ,  to  a  whole  number 
and  a  proper  fraction,  we  have  only  to  consider  how  many 


FRACTIONS.  55 

whole  ones  the  fraction  is  equal  to,  and  how  much  remains. 
Thus  J^  is  equal  to  3 ;  J^a  therefore,  is  equal  to  3^. 

Reduce  f ,  Y->  V;,  ¥,  ^,  ^,  ih  H,  ^' 

d[n  like  manner,  if  we  have  a  whole  number  and  a  fraction, 
we  may  always  reduce  it  to  an  improper  fraction. 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS. 

Suppose  we  wish  to  add  together  3^^  so  that  its  value  shall 
be  expressed  in  a  single  expression ;  we  must  change  3  to 
halves,  which  will  be  | ;  adding  ^  to  this  we  have  J  for  the 
answer. 

In  order  to  unite  separate  numbers  into  one  expression, 
they  must  be  of  the  same  kind.  We  cannot  unite  2  bushels 
and  3  pecks  in  one  expression.  It  is  still  2  bushels  and  3 
pecks,  and  we  can  make  nothing  else  of  it ;  but  if  we  change 
the  bushels  to  pecks,  making  8  pecks,  we  can  then  add  the 
3  pecks,  and  bring  it  all  into  one  expression,  11  pecks.  So 
to  unite  5f  we  must  change  the  5  to  thirds,  making  ^/,  and 
add  the  |,  making  y .  This  is  called  reducing  a  mixed,  num- 
ber to  an  improper  fraction. 

Reduce  to  an  improper  fraction  7J,  8|,  4y,  5J,  6J,  9J,  3|, 
5?,  15^,  16f,  13|,  20|,  2U. 

Supposing  we  wish  to  add  ^  to  i,  we  must  change  the  ^  to 
fourths,  making  | ;  adding  these,  we  have  |  for  the  answer. 

Add  J  to  yV-    h=T%^  A-h:V=Aj  ans. 

Add  jto^V    1+A.    ^+#iT..f+#^..  l+il 

Let  us  now  add  f  and  |.  This  question  you  perceive  has 
a  difficulty  which  the  former  ones  had  not ;  for  §  is  no  number 
of  fifths,  and  therefore  we  cannot  bring  the  fraction  into  fifths 
by  any  multiplication.  We  want  a  number  for  the  denomina- 
tor which  can  be  divided  both  by  3  and  by  5.  Now  if  you 
examine,  you  will  find  no  such  number  until  you  come  to  15. 
This,  is  of  course,  divisible  by  3  and  by  5,  for  these  are  its 
factors.  We  will  then  take  15  for  the  denominator.  This 
we  call  the  common  denominator.  Taking  now  the  fractions 
§  and  I,  and  changing  the  denominator  3  to  15,  we  see  that 
we  have  made  it  5  times  as  large  as  it  was  before ;  that  is,  we 
have  multiplied  it  by  5.  We  must  therefore  multiply  the 
numerator  by  5,  to  preserve  the  value  of  the  fraction.  The 
fraction  §  then  becomes  yj-  without  altering  its  value.     Pass- 


56  MENTAL    ARITHMETIC. 

ing  now  to  the  second  fraction,  |,  we  see  that,  in  changing  the 
denominator  to  15,  we  have  multiplied  it  by  3  ;  we  must  there- 
fore multiply  its  numerator  by  3.  This  will  make  the  fraction 
-}-|.  The  two  fractions  will  stand,  then,  if +t|-,  which  added 
together  are  y|==lTV- 

TO  FIND  A  COMMON  DENOMINATOR. 

We  can  always  obtain  a  common  denominator,  by  multi- 
plying all  the  denominators  together ;  then,  for  the  numera- 
tors, consider,  in  the  case  of  each  fraction,  what  its  denomina- 
tor has  been  multiplied  by,  in  order  to  change  it  to  the  com- 
mon denominator,  and  multiply  the  numerator  by  the  same 
number.  Thus  each  fraction  will  have  had  its  numerator  and 
its  denominator  multiplied  by  the  same  number,  and  so  its 
value  will  not  be  changed. 

What  is  the  value  of  J+f?  of  f+f?  of  f+J?  of  i+f  ? 
ofJ+f?off+§?off+|?of|+|? 

Supposing  we  wish  to  add  the  fractions  f  and  §.  We  can 
proceed  as  above,  and  with  the  common  denominator,  24,  the 
fractions  will  be  ^|-t-||.  But  we  need  not  employ  so  large  a 
denominator  as  24.  We  seek  the  smallest  denominator  that 
shall  contain  both  4  and  6  as  a  factor.  If  now  we  separate 
4  and  6  into  their  prime  factors,  we  shall  find  the  factor  2 
belonging  both  to  4  and  to  6  ;  thus,  2X2,  2X3.  Now  one 
of  these  may  be  cancelled,  and  we  shall  still  have  2X2  for  the 
number  4,  and  2X3  for  the  number  6.  Multiplying  the  fac- 
tors which  remain,  2X2X3,  we  have  12  for  the  smallest 
common  denominator. 

From  this  we  see,  that,  when  both  the  denominators  con- 
tain the  same  factor,  we  may  reject  it  from  one  of  them,  and 
multiply  together  the  factors  that  remain. 

Add  I  to  y^--  Here  2  X  2  is  common  to  both  denominators, 
rejecting  it  in  one,  and  multiplying,  we  obtain  24  for  the  least 
common  denominator. 

Add  i\  to  3jy.  Here  3X3  is  common  to  both  denomina- 
tors, rejecting  it  in  one,  and  multiplying  what  remains,  we  have 
54  for  the  least  common  denominator. 

Add^Jto^V     Add  I  to  ^V     Add^to^jV 

When  more  fraction?  than  two  are  to  be  added  it  is  often 
most  convenient  to  add  two  together  first,  and  then  add  a  third 
to  the  sum  of  these,  and  so  on. 


FRACTIONS.  •  67 

Add  l+i+f  •  -f'irst  add  f  and  ^,  which  equal  f .  Next, 
ffl ;  ^=\h  and  f=f^;  ig+^:H=i|=lT^^,  Ans. 

Add  i+f+i^d-  First  add  ^  and  ^jj ;  then  to  the  sum  of 
these  add  f . 

AddJ+,V+f-     Addf+I+T^.     Addi+f+f. 

Addf;j+^-.     AddA+f.     AddT\+^+^. 

From  ^1^  subtract  /^.     From  |  sub.  -^^.     From  -j^^  sub.  /^j. 

From  If  sub.  |.     From  f  §  sub.  y^^.     From  f  ^  sub.  ^^. 

Miscellaneous  Examples. 

1.  A  man  spends  y  of  a  dollar  in  a  day;  what  part  of  a 
dollar  will  he  spend  in  5  days  ?  How  much  -will  he  spend  in 
9  days?     How  much  in  11  days? 

2.  A  man  earns  |  of  a  dollar  in  a  day  ;  how  much  will  he 
earn  in  half  a  day  ?  How  much  in  |:  of  a  day  ?  How  much 
in  "I  of  a  day  ? 

Here  consider  whether  you  can  divide  the  numerator. 

3.  A  man  earns  J  of  a  dollar  in  a  day ;  how  much  can  he 
earn  in  half  a  day  ?  How  much  in  ^  of  a  day  ?  How  much 
in  J  of  a  day  ? 

Consider  whether  you  can  divide  the  numerator ;  and  if 
you  cannot,  what  you  must  do. 

4.  A  vessel  filled  with  water  leaks  so  that  f  of  its  contents 
will  leak  out  in  a  week ;  at  this  rate,  what  part  will  leak  out 
in  a  day  ? 

What  is  I  of  I? 

5.  If  a  team  ploughs  y  of  an  acre  in  6  hours,  how  much  will 
it  plough  in  one  hour?     How  much  in  3  hours? 

What  is  J  of  I     What  is  ^  of  ^  ? 

6.  If  a  horse  runs  |  of  a  mile  in  one  minute,  how  far  will 
he  run  in  |  of  a  minute  ? 

How  far  will  he  run  in  ^  of  a  minute  ? 
What  is  I  of  J  ?     What  is  f  of  J  ? 

7.  A  man  has  J  of  a  dollar,  which  he  wishes  to  distribute 
equally  among  several  persons,  giving  -^^  of  a  dollar  to  each ; 
how  many  can  receive  this  sum,  and  what  will  be  the  re- 
mainder? 

How  many  times  is  xV  contained  in7?  -^in??  -T^inf? 
How  many  times  is  -^  contained  in  4  ?  -^^  in  4?  ^f^  in  ^? 
How  many  times  is  ^  contained  in  6?     ^in6?     ^inf? 


58  MENTAL    ARITHMETIC. 

8.  A  man  gave  y  of  a  bushel  of  oats  to  some  horses,  giv- 
ing to  each  |^  of  a  bushel;  to  how  many  did  he  give  it?  and 
what  was  the  remainder  ?  '  ^ 

How  many  times  will  y^^  go  in  5  ?  In  ^  ?  How  many 
limes  will  y\  go  in  f  ? 

9.  A  man  has  J  of  a  dollar ;  he  gives  ^  of  a.  dollar  to  one 
person,  and  f  of  a  dollar  to  a  second,  what  part  of  a  dollar 
has  he  left  ? 

How  many  cents  had  he  at  first  ?  How  many  cents  did  he 
give  away  ?     How  many  cents  had  he  Jeft  ? 

10.  If  13  pounds  of  figs  cost  |  of  a  dollar,  what  is  that  a 
pound  ? 

11.  If  5|-  lbs.  of  figs  cost  y\  of  a  dollar,  what  is  that  a 
pound  ?     Find  first  what  one  half  pound  will  cost. 

12.  If  f  of  a  cwt.  of  iron  cost  4J  dollars,  what  will  a  hun- 
dred weight  cost  ? 

13.  If  34^  lbs.  of  tea  cost  11|  dollars,  what  will  1  pound 
C06t? 

Here  you  find  ^^  pounds  cost  \^  of  a  dollar :  therefore  69 
pounds  must  cost  \^  of  a  dollar. 

14.  If  §  of  a  barrel  of  flour  cost  3|  dollars,  what  is  that  a 
barrel  ? 

15.  If  wood  is  5 1-  dollars  a  cord,  what  will  y^^  of  a  cord 
cost  ?    What  will  4J  cords  cost  ? 

16.  If  33  J  gals,  of  molasses  cost  11|  dollars,  what  is  that 
a  gallon? 

17.  If  31 J  gals,  of  vinegar  cost  4|  dollars,  what  is  that  a 
gallon  ? 

18.  If  a  bottle  of  wine  containing  IJ  pints  cost  f  of  a  dol- 
lar, what  would  a  bari'el  of  wine  come  to  at  that  rate  ? 

19.  In  a  pile  of  wood  there  a  13|^  cords  ;  how  many  loads 
of  f  of  a  cord  each  are  there  in  the  pile  ? 

20.  How  many  times  will  2^  go  in  7|  ?     In  9J  ?     In  11  ? 

21.  How  many  loaves,  of  8^  oz.  of  flour  each,  can  be  made 
from  7  pounds  of  flour  ? 

22.  If  a  family  consume  3  J  pounds  of  flour  a  day,  how  long 
will  a  barrel  of  flour,  that  is  196  pounds,  last  them? 

How  long  will  it  last  if  they  consume  2|  lbs.  a  day  ? 

23.  If  a  barrel  of  flour  last  a  family  40  days,  how  long  will 
14  pounds  last  them  ? 

24.  A  garrison  of  100  men  is  allowed  12  oz.  of  flour  a 
day  to  each  man ;  how  long  will  10  barrels  last  them  ? 


FRACTIONS.  59 

25.  Two  men  hire  a  horse,  a  week,  for  5  dollars  ;  one  trav- 
els with  him  30  miles,  the  ether  45  miles ;  what  oflght  each  to 
pay? 

26.  Two  men  hire  a  pasture  in  common  for  $4,80 ;  one 
pastures  his  horse  in  it  7^  weeks ;  the  other  pastured  his  horse 
9  weeks  ;  what  ought  each  to  pay  ? 

27.  A  boy  "bought  3  doz.  of  oranges  for  372-  cents,  and  sold 
them  for  1^  cents  apiece  ;  what  did  he  gain  ? 

28.  A  man  bought  7  yds.  of  cloth  for  16  dollars,  and  sold 
it  for  3  dollars  a  yard  ;  what  did  he  gain  on  each  yard  ? 

29.  A  man  worth  1690  dollars,  left  f  of  his  property  to  his 
wife  ;  how  much  did  she  "receive  ?  The  remainder  he  divided 
equally  among  3  sons  ;  what  did  each  one  receive  ? 

30.  A  man  bequeathed  his  estate  of  14,000  dollars,  one 
third  to  his  wife,  and  the  remainder  to  be  divided  equally 
among  four  sons ;  what  did  the  wife  and  what  did  each  son 
receive. 

31.  In  an  orchard  one  third  of  the  trees  bear  apples,  two 
fifths  as  many  bear  plums,  and  the  rest  bear  cherries ;  what 
portion  of  the  trees  bear  plums  ?  What  portion  Jbear  cher- 
ries ?  The  number  of  cherry  trees  is  40  ;  what  is  the  whole 
number  of  trees  in  the  orchard  ? 

32.  What  is  f  of  549  ?     What  is  |  of  374  ? 

33.  What  is  i  of  175^  ?     What  is  |  of  198  ? 

34.  What  is  ^  of  I  of  1640  ?     What  is  §  of  972  ? 

35.  If  2  barrels  of  flour  cost  111  dollars,  what  will  17  bar- 
rels cost  ?     What  vAll  22^  barrels  cost  ? 

36.  If  2^  cords  of  wood  cost  15  dollars,  what  will  68f  cords 
cost  ?     What  will  200  cords  ? 

37.  If  a  horse  eat  2^  tons  of  hay  in  30  weeks,  what  part 
of  a  ton  will  he  eat  in  1  week  ? 

38.  What  is  the  cost  of  23  J  yds.  of  cloth  at  J  of  a  dollar  a 
yard  ? 

39.  What  is  the  cost  of  31 J  gallons  of  molasses  at  y^  of  a 
dollar  a  gallon  ? 

40.  A  grocer  drew  from  a  cask  containing  31J^  gallons,  J 
of  its  contents.  Now  how  much  did  he  draw  out?  How 
much  remained  ? 


60  MENTAL   ARITHMETIC. 

SECTION    X. 

THE  LEAST  COMMON  MULTIPLE. 

The  method  stated  in  the  foregoing  section  for  finding  the 
smallest  common  denominator,  serves  to  introduce  a  topic 
which  requires  some  more  extended  and  careful  study. 

It  often  becomes  desirable  to  ascertain,  with  respect  to  sev- 
eral numbers,  what  number  the^e  is  which  contains  them  all 
in  itself  as  factors.  A  number  which  contains  another  num- 
ber as  a  factor  of  itself  is  a  multiple  of  that  number.  Thus 
6  is  a  multiple  of  2,  and  also  of  3.  A  number  which  contains 
several  numbers  as  factors  of  itself,  is  a  common  multiple  of 
those  numbers.     Thus  12  is  a  common  multiple  of  2  and  3. 

The  smallest  number  which  contains  several  numbers  as 
factors  of  itself,  is  the  least  common  multiple  of  those  numbers. 
Thus,  though  12  is  a  common  multiple  of  2  and  3,  it  is  not 
the  least  common  multiple ;  for  6  contains  them  both  as  its 
factors  ;  6  is  therefore  a  smaller  common  multiple  of  2  and  3 
than  12  is ;  and  as  no  number  smaller  than  6  does  contain  2 
and  3  as  its  factors,  6  is  the  smallest  common  multiple  of  2 
and  3. 

Suppose  now  we  wish  to  find  the  smallest  common  multiple 
of  3  and  5.  The  number,  it  is  clear,  must  he  a  certain  num- 
ber of  3s,  and  also  a  certain  number  of  5s.  Now  bj  multiply- 
ing 3  and  5  together  we  evidently  obtain  such  a  number  ;  for 
it  will  be  3  times  5,  and  it  will  be  5  times  3.  Multiplying 
the  two  numbers  together  then,  will  always  give  their  com- 
mon multiple.  The  next  question  is,  will  this  product  of  the 
two  numbers  be  their  least  common  multiple  ?  This  will  de- 
pend on  the  character  of  the  numbers.  If  the  numbers  are 
prime  to  each  other  their  product  will  be  their  least  common 
multiple.  For  example,  in  the  numbers  3  and  5,  if  we  take 
any  number  of  5s  less  than  3,  as  2X5,  the  factor  3  has  disap- 
peared, and  the  number  is  no  longer  a  multiple  of  3.  If  we 
take  any  number  of  3's  less  than  5,  as  4X3,  the  factor  5  has 
disappeared,  and  the  number  is  no  longer  multiple  of  5.  The 
product,  therefore,  of  numbers  prime  to  each  other,  is  their 
least  common  multiple.  In  the  above  example,  the  numbers 
of  3  and  5  were  prime  in  themselves,  and  not  merely  prime 
to  each  other.     To  make  the  principle  more  clear,  we  will 


FRACTIONS.  61 

take  two  numbers  that  are  not  prime  in  themselves,  but  are 
only  prime  to  each  other. 

What  is  the  least  common  multiple  of  8  and  9  ?  Multiply- 
ing them  together  we  Jiave  72.  72  is,  then,  a  common  multi- 
ple of  8  and  9.  The  question  is,  is  it  their  smallest  common 
multiple  ?  Writing  the  numbers  with  their  factors  they  are 
2X2X2  and  3X3.  Now  if  we  erase  one  of  the  2's  we  have 
no  longer  the  factors  of  8..  and  the  product  of  the  factors  will 
not  be  divisible  by  8.  In  the-  same  way,  if  we  erase  one  of 
the  3's  the  product  will  not  be  divisible  by  9. 

If,  then,  the  numbers  are  either  prime,  or  prime  to  each 
other,  the  product  is  their  least  common  multiple. 

Next  let  us  inquire,  what  is  the  least  common  multiple  of 
4  and  6  ?  Their  product  is  24,  but  this  is  evidently  not  their 
least  common  multiple,  for  12  contains  both  4  and  6  as  fac- 
tors. To  show  why  it  is,  that  in  this  case,  something  less 
than  the  product  of  the  numbers  is  their  least  common  _  multi- 
ple, we  will  express  each  by  its  factors,  thus,  2X2,  2X3. 
Now  it  is  clear  that  any  number  of  times  which  you  take 
2X2  as  a  factor  will  be  a  multiple  of  2X2.  If  then  we 
^hrow  out  the  2  in  the  2X3,  and  multiply  by  the  remaining 
3,  the  product  will  be  a  multiple  of  2X2,  or  4.  Looking 
now  at  the  2X3,  or  6,  it  is  evident  that  any  number  of  times 
which  you  may  take  that  as  a  factor  will  be  a  multiple  of 
2X3.  But  the  2  we  may  take  from  the  2X2,  throwing 
away  that  in  the  2X3;  this  leaves  us  to  multiply  the 
2X3  by  2  ;  as  we  before  multiplied  the  2X2  by  3,  making 
12  as  the  least  common  multiple.  The  rule,  therefore,  is : 
Retain  of  each  prime  factor  the  highest  power  which  appears 
in  any  of  the  given  numbers ;  erase  the  rest,  and  multiply  to- 
gether what  then  remain. 

Find  the  least  common  multiple  of  8,  24  and  36.  Ex- 
pressed by  the  factors  they  are  2X2X2.  2X2X2X3. 
2X2X3X3.  Now  2X2X2  is  common  to  8  and  24;  it  may 
be  thrown  out  of  the  latter,  leaving  only  3.  Examining  again 
you  observe  that  2X2  is  common  to  8  and  36;  we  throw 
this  out  of  36,  leaving  3X3.  Finally  3,  we  find,  is  common  to 
24  and  36  ;  throwing  this  out  of  24,  we  find  the  numbers 
appear  as  follows :  2X2X2.  ^X^X^X0.  ^X^X3X3. 
These  multiplied  together  give  for  the  least  common  multiple, 
72.  This  conforms  to  the  rule;  for  2X2X2  is  the  highest 
power  of  the  factor  2,  and  3X3  of  the 'factor  3.  What  is  the 
6 


62  MENTAL    ARITHMETIC. 

least  common  multiple  of  24,  60  and  100  ?  These  factors  are 
2X2X2X3;  2X2X3X5 ;  2X2X5x5.  We  see  that 
2  X  2  is  common  to  them  all ;  expunge  it  in  the  second  and 
third  number.  Next,  3  is  common  to  the  1st  and  2d ;  expunge 
it  in  the  2d.  Lastly,  5  is  common  to  the  2d  and  3d ;  expunge 
it  in  the  2d,  and  the  numbers  will  stand,  2X2X2X3. 
^X^X'^X$'  ^X^XoX^-  These  multiplied  together, 
give  600. 

To  multiply  these  most  easily,  first  take  2X2X5X5=100 ; 
then  the  remaining  factors,  2x3,  multiplied  by  100,  give  600. 

What  is  the  least  common  multiple  of  24,  40  and  72  ? 

What  is  the  least  common  multiple  of  18,  54,  81  ? 

What  is  the  least  common  multiple  of  15,  4,  7  ?  of  15,  40, 
27?  of  16,  14,  6?  of  60,  12,  18? 

From  the  foregoing  reasoning  and  examples  you  will  per- 
ceive that  the  least  common  multiple  of  several  numbers  is 
the  product  of  all  their  prime  factors,  each  taken  in  the  high- 
est power  in  which  it  appears  in  any  of  the  numbers. 


SECTION    XI. 

PRACTICAL  QUESTIONS. 

1.  What  part  of  a  shilling  is  1  penny  ?  2  pence  ?  3  pence  ? 
4  pence  ?  5  pence  ?  6  pence  ?  7  pence  ? 

2.  What  part  of  a  penny  are  2  farthings  ?    3  farthings  ? 
4  farthings  ?  5  farthings  ?  6  farthings  ?  8  farthings  ? 

3.  What  part  of  a  shilling  is  1  farthing  ?    2  farthings  ?  3 
farthings  ? 

What  part  of  a  shilling  is  1  penny  and  1  farthing  ?  1  pen- 
ny, 2  farthings  ?  3d  3  qrs.  ?  4d  2  qrs.  ?  6d  1  qr.  ?    9d  2  qrs.  ? 

4.  What  part  of  a  pound  is  1  shilling  ?    2  s.?    3  s.?    5  s.? 
Is.  Id.  ?  2s.  Id.  ?  4s.  3d.  ?  5s.  6d.?  7s.  9d.  ?  3s.  8d.  ? 

5.  What  part  of  a  pound  is   1  farthing?    2  qrs.?  3  qrs.  ? 
2d.  3  qrs.  ?  5d.  2  qrs.  ?  Is.  Id.  1  qr.  ?  6s.  7d.  3  qrs.  ? 

6.  What  part  of  a  pound  avoirdupois  is   2  oz.  ?    3  oz.  ? 
4  oz.  ?  5  oz.  ?  6  oz.  ?  7  oz.  ?  8  oz.  ?  9  oz.  ?  10  oz.  ? 


F][lACTIONS.  '  63 

7.  What  part  of  one  ounce  is  one  dram  ?  What  part  of  1 
pound  is  one  dram  ?  2  drs.  ?  3  drs.  ?  1  oz.  1  dr.  ?  1  oz. 
2  drs.  ?  2  oz.  4  drs.  ?  3  oz.  6  drs.  ?  8  oz.  3  drs.  ?  9  oz.  1 1  drs.  ? 

8.  Yv^'hat  part  of  a  pound  is  -^^  of  an  oz.  ?  -^-^  of  an  oz.  ? 
What  part  of  a*  pound  is  ^  an  oz.  ?    2^  oz.  ?    of  3^  oz.  ? 

4^oz.? 

9.  What  part  of  a  pound  Troy  is  1  dwt.  ?  5  dwt.  ?  6  dwt.  ? 
9  dwt.  ?  11  dwt.  ?  10  dwt.  ?    1  oz.  1  dwt.  ?  3  oz.  4  dwt.  ? 

What  part  of  oz.  Troy  is  1  dwt.  ?  3  dwt.  1  gr.  ?  4  dwt.  6  gr.  ? 
7  dwt.  3  grs.  ?  8  dwt.  9  grs.  ?  10  dwt.  ?    12  dwt.  ?    16  dwt.  ? 

10.  What  part  of  an  ell  English  is  1  qr.  of  a  yard  ?  2 
qrs.  ?  3  qrs.  ?     What  part  of  a  qr.  is  1  nail  ?    3  nails  ? 

11.  What  part  of  a  yd.  is  1  qr.  1  nail  ?  2  qrs.  3  n.  ?    3  qrs. 

2  n.?     What  part  of  an  ell  English  is  3  nails  ?    1  qr.  3  n.  ?  4 
qrs.  In.? 

12.  What  part  of  a  yd.  is  1  inch?  4  inches?  7  inches? 
9  inches  ?     What  part  of  a  yard  is  1  qr.  2  in.  ?     2  qrs.  3  in.  ? 

3  qrs.  1  in.  ? 

13.  From  a  vessel  containing  3  gallons  of  wine,  3  gills 
leaked  out ;  what  part  of  a  gallon  leaked  out  ?  What  part 
of  a  gallon  remained  ? 

14.  From  a  barrel  full  of  wine  7  quarts  were  drawn ;  how 
many  quarts  remained  ?  What  part  of  the  barrel  had  been 
drawn  out  ?     What  part  of  the  barrel  had  remained  ? 

15.  If  f  of  a  barrel  of  beer  be  divided  into  4  equal  parts, 
what  part  of  a  barrel  will  each  of  the  parts  be  ?  How  many 
gallons  will  each  part  be  ? 

16.  If  one  quart  be  taken  from  a  barrel  full  of  beer,  what 
part  of  a  barrel  will  remain  ?  If  3  pints  be  taken  out,  what 
part  will  remain  ?  If  7^  gallons  be  taken  out,  what  part  of 
a  barrel  is  taken  out  ?     What  part  of  a  barrel  remains  ? 

17.  A  man  distributed  7^  gallons  of  milk  among  5  persons ; 
what  part  of  a  gallon  did  he  give  to  each  ? 

18.  If  you  have  3^  gallons  of  milk,  and  distribute  it  to 
some  poor  persons,  giving  f  of  a  gallon  to  each,  how  many 
persons  will  you  give  it  to  ?     How  much  will  remain  ? 

19.  What  part  of  1  foot  is  1^  in.?  2iin.?  5^  in.?  e^in.? 
8|in.?     9iin.?     lOfin.?     11^  in.? 

20.  What  part  of  a  yard  is  2  inches  ?  3^  inches  ?  14  in.  ? 
5iin.?     6iin.?     17^  in.?     24^  in.? 

21.  What  part  of  a  rod  is  ^  a  foot?     1^  feet?     2^  feet? 

4  ft,  8  in,?     6  ft  7  in.?     10  ft.  5  in.? 


64  MENTAL   ARITHMETIC). 

22.  What  part  of  3  rods  is  |  a  foot?  1  foot?  3^  feet? 
What  part  of  a  furlong  are  2^  rods  ?  5^  rods  ? 

23.  What  fraction  of  a  foot  is  ^  of  a  yard  ?  ^  of  a  yd.  ? 
What  fraction  of  a  foot  is  -^  of  a  rod  ?  y^/of  a  rod  ?  ^  of 
a  rod  ? 

24.  A  man  measured  the  length  of  his  barn  with  a  stick 
half  a  yard  long,  arid  found  the  barn  31^  times  the  length  of 
his  stick ;  how  long  was  it  ? 

25.  A  carpenter  is  cutting  up  a  board  17^  feet  in  length, 
into  pieces  2^  feet  long  ;  how  many  pieces  will  there  be,  and 
how  long  will  be  the  piece  that  remains  ? 

26.  A  man  measures  a  piece  of  fence  with  a  pole  9^  feet 
long;  the  fence  is  15 1  times  the  length  of  the  pole;  how 
many  rods  is  it  in  length  ? 

27.  What  part  of  a  peck  is  -^jj  of  a  bushel  ?. 

What  part  of  a  gallon  are  -5^  of  a  peck  ?  f  of  a  peck  ? 
What  part  of  a  quart  is  ^^^  of  a  peck  ?  /^^  of  a  peck  ? 
What  part  of  a  quart  are  ^-^  of  a  bushel  ?  ^^  of  a  bushel  ? 

28.  What  part  of  a  peck  is  -^  of  a  bush.  ?  |  of  a  bush.  ? 
f  of  a  bush.  ?  ^  of  a  bush.  ?  |-  of  a  bush.  ? 

29.  Two  men  bought  a  lot  of  standing  wood  in  company, 
for  11  dollars;  one  cut  off  2  cords,  the  other  1  cord;  what 
ought  each  to  pay  ? 

30.  Two  boys  bought  the  chesnuts  on  a  tree  for  50  cents ; 
one  had  11  quarts,  the  other  6  quarts  and  1  pint;  what  ought 
each  to  pay  ? 

31.  Three  men  bought  a  piece  of  cloth  for  24  dollars? 
the  first  took  2|-  yds.,  the  second  the  same  quantity,  and  on 
measuring  the  remainder  it  was  found  to  be  3  yards  ;  what 
ought  each  to  pay  ? 

32.  Two  men  hire  a  horse  for  a  month  for  12  dollars  ;  one 
travels  200  miles  with  the  horse,  the  other  150 ;  how  much 
should  each  pay  ? 


JH 


FRACTIONS.  65 

SECTION    XII. 

DECIMAL    FRACTIONS. 
[3ee  Numeration,  Part  IT.  J 

In  tlie  calculations  in  common  fractions,  a  great  inconve- 
nience arises  from  their  irregularity.  There  is  no  law  regu- 
lating the  magnitude  of  either  of  the  terms.  The  denomina- 
tor may  be  in  any  ratio  whatever  to  the  numerator.  From 
seeing  one  you  can  make  no  inference  at  all  respecting  the 
magnitude  of  the  other.  In  calculations  of  addition,  it  is 
often  more  than  half  the  work  to  bring  the  fractions  into  a 
common  denomination. 

Now  it  is  evident  that  if  fractions  could  be  written  in  the 
same  manner  as  whole  numbers,  that  is,  increasing  in  a  ten- 
fold rate  as  you  advance  to  the  left,  and  decreasing  in  a 
ten  fold  rate  as  you  advance  to  the  right,  an  immense  gain 
would  be  made  in  the  convenience  of  calculating  them.  Op- 
erations in  fractions  would  then  be  just  as  ea^  as  operations 
in  whole  numbers.  Now  this  advantage  is  gained  in  decimal 
fractions.  They  are  brought  under  the  same  law  as  whole 
numbers.  Let  us  observe  the  manner  in  which  whole  num- 
bers are  written.  Take  the  number  222  ;  the  right  hand 
figure  signifies  two  units,  the  next  two  tens,  the  next  two  hun- 
dreds; just  as  if  it  were  written  in  this  manner,  2Xl00-f- 
2X10+2:  two  multiplied  by  100  plus  two  multiplied  by  10, 
plus  two;  making  two  hundred  and  twenty-two.  But  thi 
cumbersome  method  of  writing  is  unnecessary,  because  the 
law  of  notation  determines  what  number  the  figures  in  each 
place  shall  be  multiplied  by.  It  must  not  be  forgotten  that 
the  figure  2  in  the  above  example  in  no  case  signifies  of  itself 
more  than"^  two.  It  is  the  place  it  occupies  that  gives  it  the 
higher  value  of  tens  or  hundreds. 

Now  it  would  evidently  be  a  great  convenience  if  we  could 
reduce  fractions  to  the  same  law,  so  that  they  would,  like 
whole  numbers,  decrease  in  a  decimal  ratio,  in  advancing 
from  the  left  to  the  right.  To  show  this  regularity  to  the 
eyf,  we  will  write  the  following  numbers  :  two  multiplied  by 
1000,  two  multiplied  by  100,  two  multiplied  by  10,  two  units, 
two  divided  by  10,  two  divided  by  100,  and  two  divided  by 
1000.  Written  in  full  they  would  stand  thus:  2X1000+ 
2  X 100+2  X  10+2+A+Tf  tt+t/^^. 
6* 


66  ■      MENTAL   ARITHMETIC. 

But  we  have  seen  that  we  may  write  the  whole  numbers 
without  the  multipliers,  thus,  2222,  because  we  know  from  the 
place  each,  figure  occupies  what  its  multiplier  must  be.  Just 
so  we  can  write  fractions  without  the  denominators,  provided 
we  know,  from  the  place  of  the  numerator,  what  the  denomi- 
nator must  be.  Thus  the  whole  of  the  above  series  may  be 
written  as  follows ;  2222.222.  A  decimal,  therefore,  is  the 
numerator  of  a  fraction,  whose  denominator  is  never  written, 
but  is  always  understood  to  be  1,  with  as  many  ciphers  as 
there  are  places  in  the  decimal. 

You  observe  that,  in  writing  the  series  given  above,  there 
is  a  period  placed  at  the  right  hand  of  the  whole  numbers, 
separating  the  unit  figure  from  that  of  tenths.  The  period 
must  never  be  omitted  when  there  are  fractions,  for  it  enables 
you  to  determine  the  value  of  each  figure  in  the  sum.  In- 
stead of  reading  .22  two  tenths  and  2  hundredths,  we  may 
call  it  22  hundredths,  which  is  more  convenient  and  amounts 
to  the  same ;  for  2  tenths  is  equal  to  20  hundredths ;  so 
.222  is  two  hundred  and  twenty-two  thousandths.  So,  in  all 
cases,  read  the  decifaal  numbers  as  whole  numbers,  and  for 
their  denominator  take  1  with  as  many  ciphers  as  there  are 
places  in  the  written  decimals. 

In  all  your  study  of  decimals,  be  careful  not  to  confound 
the  words  which  express  frictions  with  the  similar  words 
which  express  whole  numbers  ;  as  tenths  with  tens,  hundredths 
with  hundreds.  The  following  questions  will  aid  you  in  fixing 
this  distinction  clearly  in  mind. 

1.  How  many  tenths  are  equal  to  ten  whole  ones  ? 

2.  How  many  tenths  are  equal  to  two  and  a  half  whole 
ones? 

3.  How  many  hundredths  are  equal  to  three  and-  a  quarter 
whole  ones  ? 

4.  How  many  hundredths  are  equal  to  one  hundred  whole 
ones  ? 

5.  How  many  thousands  are  equal  to  ten  whole  ones  ? 

6.  In  fifteen  whole  ones  how  many  tenths  ?  How  many 
hundredths  ? 

7.  In  seventy-five  hundredths  how  many  tenths  ? 

8.  In  three  tenths  how  many  hundredths  ? 

9.  In  six  tenths  how  many  thousandths  ? 

Thus,  you  observe,  fractions  have  been  brought  under  the 
same  law  that  regulates  the  writing  of  whole  numbers.    .They 


ADDITION   AND    SUBTRACTION    OF   DECIMALS.  67 

may  now  be  added,  subtracted,  multiplied,  and  divided,  like 
whole  numbers.  But  in  doing  this  it  is  important  to 
determine  the  place  of  the  period  that  separates  the  whole 
numbers  from  the. fractional  part  of  the  sum.  Where  must 
the  period  be  placed  in  the  answer  ? 


ADDITION  AND  SUBTKACTION  OF  DECIMALS. 

Let  us  first  observe  how  important  it  is  that  the  rule  in  this 
case  be  entirely  correct.  If  I  have  this  number,  32.5,  to  write, 
and  by  any  mistake  I  should  write  it  3.25,  it  would  denote  a 
quantity  only  one  tenth  as  great  as  it  should  be ;  or,  if  I  should 
write  325.  it  would  denote  a  quantity  ten  times  greater  than  it 
should  be.  Moving  the  period  one  place  to  the  right,  makes 
the  number  ten  times  as  great  as  it  was  before,  for  tens  be- 
come hundreds,  and  hundreds,  thousands  ;  and  each  figure  ten 
times  as  great  as  before.  So,  by  moving  the  period  one  place 
to  the  left,  the  number  becomes  just  one  tenth  what  it  was 
before.  Removing  the  period  two  places  from  its  true  place, 
makes  the  number  100  times  larger  or  smaller  than  it  should 
be,  according  as  you  remove  it  to  the  right  or  the  left.  Hence 
you  may  see  that  in  order  to  multiply  a  number  that  has 
decimals,  by  10,  you  have  only  to  remove  the  period  one  place 
to  the  right;  to  multiply  by  100,  remove  it  two  places,  and  so 
on.  To  divide  by  10,  remdve  the  period  one  place  to  the 
left ;  to  divide  by  100,  remove  it  two  places,  and  so  on.  From 
the  above  you  may  see  the  importance  of  being  perfectly  ac- 
curate in  fixing  the  place  of  the  decimal  in  the  answer  to  any 
question. 

We  will  begin  with  addition.  Add  4.46  to  3.21.  Here  you 
observe  the  two  whole  numbers  make  7,  and  46  hundredths 
added  to  21  hundredths  make  67  hundredths  :  the  answer, 
then,  must  be  7.  67,  having  two  decimal  places.  Add  6.  8  to 
5.  23.  The  3  hundredths  must  evidently  stand  alone,  since 
there  is  nothing  like  it  to  add  to  it ;  2  tenths  added  to  8  tenths 
make  10  tenths,  or  one  whole  one  ;  this  we  carry  to  the  5, 
which  gives  us  for  the  answer,  12.  03.  This  will  serve  to 
suggest  the  rule  for  placing  the  period  in  the  answer  to  ques- 
tions in  addition.  The  number  of  decimal  places  in  the 
answer  must  be  as  great  as  can  be  found  in  any  one  of  the 
numbers  to  be  added. 


68  MENTAL    ARITHMETIC. 

The  same  rule  holds  in  subtraction.  Take  for  illustration 
the  numbers  given  in  the  second  example  of  addition.  From 
6.  8  subtract  5.  23.  Now  as  in  the  minuend  there  are  no 
hundredths,  we  must  borrow  10  in  this  place,  and  we  shall 
have  a  remainder  of  7  hundredths ;  adding  1  tenth  to  the  sub- 
trahend, to  compensate  for  the  10  hundredths  added  to  the 
minuend,  we  have  in  the  place  of  tenths  a  remainder  of  5  ; 
finally,  in  the  place  of  units  we  subtract  5  from  6  :  the  answer 
is  1.  57.  In  performing  this  operation,  you  may,  if  you  please, 
call  the  8  tenths  80  hundredths ;  then  23  hundredths  from  80 
hundredths  leaves  57  hundredths.  By  performing  slowly 
and  with  care  examples  of  your  own  selection,  you  will  see 
the  verification  of  the  rule  given  above,  both  for  addition  and 
subtraction. 

Add  2.4  to  3.8.  Add  .6  to  1.3.  Add  .4  to  .3.  Add  .37 
to  .25.     Add  3.7  to  .24.     Add  1.08  to  .05. 

From  4.6  subtract  2.4.  From  7.1  subtract  6.4.  From  .18 
subtract  .13.     From  4.5  subtract  .6. 

In  these  examples  each  step  should  be  explained  by  the 
pupil  as  he  performs  it. 


MULTIPLICATION  OF  DECIMALS. 

The  rule  in  multiplication  we  shall  find  to  be  different  from 
the  above. 

1.  First,  we  will  multiply  2.4  by  3.  If  we  regard  the 
multiplicand  as  a  whole  number,  the  answer  Avill  be  72.  But 
by  regarding''  the  multiplicand  as  a  whole  number,  —  as  24 
instead  of  2  and  4  tenths,  —  we  regarded  it  ten  times  greater 
than  it  really  is ;  the  answer,  therefore,  is  ten  times  too  great. 
Instead  of  72  it  must  be  7.2. 

2.  Multiply  6.2  by  3.4.  By  regarding  both  as  whole 
numbers  we  obtain  the  answer  2108.  Now  in  calling  the 
multiplicand  62  instead  of  6.2  we  treated  it  as  10  times 
greater  than  it  is.  The  answer  must  therefore  be  10  times 
too  great,  even  if  the  multiplier  were  a  whole  number.  Wp 
must  therefore  divide  it  by  10,  or  write  210.8.  But  the  mul- 
tipHer  also  is  10  times  too  great;  the  answer  must  therefore 
be  divided  again  by  10,  in  order  to  bring  it  right.  Thus  the 
answer  will,  stand  21.08. 

3.  Again  ;  multiply  .62  by  3.4.     Here  we  obtain  the  same 


i 

DIVISION    OF   DECIMALS.  69 

figures  as  before,  2108  ;  but  by  treating  the  multiplicand  as  a 
whole  number,  we  regarded  it  as  100  times  too  great ;  the 
answer  therefore  must  be  divided  by  100,  or  written  21.08. 
But  the  multiplier,  calUng  it  a  whole  number,  was  taken  10 
times  greater  than  it  is ;  the  answer  must  be  again  divided  by 
10,  and  thus  it  will  stand  2.108. 

4.  Once  more ;  multiply  .62  by  .34.  The  figures  of  the 
answer  are  as  before,  2108,  but  by  regarding  both  the  factors 
as  whole  numbers,  we  take  each  100  times  greater  than  it  is  ; 
we  must  therefore  divide  by  100  to  correct  the  error  in  the 
multiplier,  and  again  by  100  to  correct  the  error  in  the  multi- 
plicand. This  will  remove  the  point  four  places  to  the  left, 
and  the  true  answer  will  be  .2108.  By  examining  these 
examples  you  will  see  that  the  pointing  in  each  case  conforms 
to  the  following  rule. 

Point  off  as  many  figures  for  decimals  in  the  ^answer  as 
there  are  decimal  places  in  both  the  factors  taken  together. 

5.  Multiply  2.7  by  .3.  6.  Multiply  .6  by  .7.  7.  Multiply 
6.  by  .7.  8.  Multiply  .02  by  .3.  9.  Multiply  .02  by  .03. 
10.  Multiply  .01  by  .01. 


DIVISION  OF  DECIMALS. 

1.  Divide  48  by  12.     Ans.  4. 

2.  Divide  4.8  by  12.  The  figure  expressing  the  answer 
is  4,  as  in  the  first  case ;  but,  observe,  the  dividend  is  only 
one  tenth  as  large  as  before  ;  the  quotient,  therefore,  is  only 
one  tenth  as  large.     Instead  of  4.  it  is  .4. 

3.  Divide  .48  by  12.  The  figure  of  the  quotient  is  still  4, 
but  as  the  dividend  is  only  one  hundredth  part  as  large  as  in 
the  first  example,  the  quotient  will  be  only  one  hundredth  part 
of  4,  or  4  hundredths,  written  thus,  .04. 

4.  Again ;  divide  48  by  1.2.  The  quotient  is  still  4,  but  we 
must  investigate  the  question  to  see  where  this  4  must  stand. 
You  observe  that  the  divisor  is  now  only  one  tenth  of  12. 
Now  if  the  divisor  is  only  one  tenth  as  great  as  it  was  before, 
you  must  consider  how  that  will  affect  the  quotient.  You  will 
perceive  on  reflection  that  as  you  diminish  the  divisor  you 
increase  the  quotient.  If  you  make  the  divisor  half  as  great, 
the  quotient  will  be  twice  as  great,  and  so  proportionally  of 
other  numbers.     Now'as,  in  this  instance,  thr^  divisor  is  one 


70  MENTAL    ARITHMETIC. 

tenth  as  great  as  before,  the  quotient  must  be  ten  times  greater. 
The  figure  4,  then,  which  is  the  quotient  figure,  instead  of 
standing  in  the  place  of  units,  as  before,  must  stand  in  tho 
place  of  tens  ;  that  is,  it  must  be  40,  the  cipher  merely  show- 
ing that  the  4  stands  in  the  place  of  tens. 

5.  Once  more  :  divide  48  by  .12.  Here  again  you  have  4 
for  the  quotient  figure,  for  you  can  have  no  other ;  but  on 
comparing  this  example  with  the  first,  you  perceive  the  divisor 
is  only  one  hundredth  part  as  great ;  the  quotient  must  there- 
fore be  one  hundred  times  greater,  that  is,  it  is  400,  the 
ciphers  merely  removing  the  4  into  the  place  of  hundreds, 

On  examining  these  examples  carefully,  you  will  see  that 
each  answer  is  unquestionably  correct.  "  But  by  what  rule," 
you  ask,  "are  these  examples  wrought?"  They  are  not 
wrought  by  rule,  but  by  reasoning  on  the  numbers  themselves ; 
and  the  more  you  habituate  yourself  to  reason  in  arithmetic, 
the  less  need  you  will  have  to  depend  on  rules. 

With  this  suggestion  I  will  now  state  a  rule,  which  you  may 
at  any  time  follow,  when  you  have  not  time  to  look  into  the, 
reason  of  the  operation. 

There  must  be  as  many  decimals  in  the  quotient  as  the 
decimals  in  the  dividend  exceed  those  in  the  divisor :  when 
there  are  fewer  decimals  in  the  dividend  than  there  are  in 
the  divisor,  ciphers  must  be  added  so  as  to  make  the  number 
equal. 

We  will  now  review  the  foregoing  examples,  and  observe 
their  conformity  with  the  above  rule.  Example  1  has  no 
decimals  in  the  divisor  or  the  dividend,  therefore  none  in  the 
quotient.  Ex.  2,  the  dividend  has  one  decimal,  the '  divisor 
none ;  the  quotient  has  therefore  one.  Ex.  3,  the  dividend 
has  two  decimals,  the  divisor  none ;  the  quotient  has  two. 
Ex.  4,  the  dividend  has  none,  the  divisor  one ;  there  must 
then  be  a  cipher  added  to  the  dividend,  and  then  the  quotient 
will  be  in  whole  numbers.  Ex.  5,  the  dividend  has  none,  the 
divisor  two  ;  there  must  then  be  two  ciphers  added,  and  then 
the  quotient  will  be  in  whole  numbers. 

6.  Divide  45  by  15.  Divide  4.5  by  15.  Divide  .45  by 
15.     Divide  45  by  1.5.     Divide  45  by  .15. 

7.  Divide  66  by  11.  6.6  by  11.  .66  by  11.  66  by  1.1. 
66  by  .11. 

In  calculations  of  Federal  money,  cents  and  mills  are  re- 
garded as  decimals  ;  the  point  therefore  separating  the  whole 


ANALYSIS    OF    DECIMALS.  71 

numbers  from  the  fractions  must  be  placed  between  th#dol- 
lars  and  the  cents.  Thus  24.00  is  24  dolls. ;  2.40  is  2  dolls. 
40  cents ;  0.24  is  24  cents. 

8.  A  man  divided  $24.00  among  3  men ;  how  much  did 
each  receive  ? 

9.  A  man  divided  $2.40  among  3  men ;  how  much  did  each 
receive  ?    Divide  2.4  by  3. 

10.  A  man  divided  $0.24  among  3  men ;  how  much  did 
each  receive  ?     Divide  $0.24  by  3. 

11.  A  man  divided  36  dollars  among  4  persons;  how  much 
did  each  receive?     Divide  36  by  4. 

12.  A  man  divided  $3.60  among  4  persons  ;  how  much  did 
each  receive  ?     What  is  one  fourth  of  $3.60? 

13.  A  man  divided  $0.36  among  4  men;  how  much  did 
each  receive  ?     What  is  one  fourth  of  .36  ? 


SECTION    XIII. 

REDUCTION  OF  VULGAR  FRACTIONS  TO  DECIMALS. 

We  have  now  seen  that  Decimal  Fractions  have  this  great 
advantage  over  Vulgar  Fractions,  —  that  they  conform  to  the 
same  law  of  notation  as  whole  numbers,  and  may  be  added, 
subtracted,  multiplied  and  divided  in  the  same  manner,  and 
with  the  same  ease  as  whole  numbers^  It  is  desirable,  there- 
fore, to  introduce  them  in  a  great  many  cases  instead  of 
Vulgar  Fractions.  The  next  question  that  arises,  therefore, 
is,  can  a  Vulgar  Fraction  be  changed  to  a  decimal,  ha\-ing 
the  same  value  ;  and  how  can  it  be  done  ?  Take  the  fraction 
^ ;  we  wish  to  reduce  it  to  tenths ;  or  in  other  words  to  ex- 
press it  in  tenths.  Now  we  can  change  any  number  to  tenths 
by  multiplying  it  by  10.  Thus  3  is  30  tenths,  4  is  40  tenths. 
We  will  now  take  ^  and  change  the  numerator  1  to  tenths, 
and  it  will  stand  .10  :  but  the  fraction  was  not  one,  but  one 
half  of  one ;  10  therefore  is  twice  as  great'  as  it  should  be ; 
we  must  divide  it,  therefore,  by  2  ;  that  is,  by  the  denomina- 
tor, and  it  will  be  .5.  To  reduce  a  vulgar  fraction,  then,  to  a 
decimal :  add  a  cipher  to  the  numerator,  and  divide  by  the 


72  MENTAL    ARITHMETIC. 

denominator.     If  one   cipher  is  not  enough   to   render  the 
division  complete,  add  more. 

Reduce  to  a  decimal  -^  ;  change  the  numerator  to  tenths ;  it 
will  be  .10,  but  the  quantity  to  be  reduced  to  tenths  was  not 
one,  but  one  fifth  of  one ;  10,  therefore,  is  5  times  greater  than 
it  should  be  ;  dividing  by  5,  the  answer  is  .2. 

Reduce  to  a  decin^al  the  fraction  f ,  explaining  each  step  in 
the  operation. 

Reduce  to  a  decimal  the  fraction  ^. 

Reduce  to  a  decimal  the  fraction  |. 

Reduce  to  a  decimal  the  fraction  ^. 

Reduce  to  a  decimal  the  fraction  | . 

I  will  here  direct  your  attention  to  a  fact  that  it  is  interest- , 
ing  to  notice.  If  the  denominator  of  the  vulgar  fraction  is 
one  of  the  factors  of  10,  that  is,  if  it  is  either  2  or  5,  the 
decimal  figure  will  be  as  many  times  the  other  factor  as 
there  are  units  in  the  numerator  of  the  vulgar  fraction. 
This  will  appear  self-evident  when  we  express  the  numbers 
by  their  factors.  Thus  in  obtaining  the  decimal  for  ^  we 
divide  10  by  2;  but  10  is  2X5,  therefore  in  dividing  by  2* 
we  simply  expunge  the  factor  we  divide  by,  and  leave  the 
other  :  2)  ^X5.  So  in  the  fraction  ^,  we  obtain  the  decimal 
by  dividing  10  by  5,  which  expunges  the  factor  5,  5)^X2; 
in  reducing  f  we  divide  2  X 1 0  by  5,  thus :  5)  2  X  2  X  ^,  leaving 
twice  the  factor  2  ;  in  f ,  5)  3  X  2  X  ^,  leaving  3  times  the  fac- 
tor 2;  in  I,  5)  2X2X2X^,  leaving  4  times  the  factor  2. 

2.  We  will  now  take  the  fraction  ^ ;  proceeding  as  before 
we  wish  to  divide  10  by .'4,  thus,  2X2)  2X5;  here  we  seethe 
division  cannot  be  complete,  for  the  divisor  contains  the  fac- 
tor 2  twice,  while  the  dividend  has  it  only  once.  If,  however, 
we  had  multiplied  the  original  numerator  1  by  100,  instead 
of  10,  we  should  have  had  10  twice  as  a  factor  in  the  dividend, 
and  of  course  each  factor  of  10  twice;  100  is  10X10,  and  10 
is  2X5.  It  would  have  stood  then  thus,  2X2)2X5X2X5; 
the  division  is  now  complete,  for  the  dividend  contains  the 
factor  2  as  many  times  as  the  divisor  has  it.  Expunging 
these  we  have  remaining  the  factor  5  taken  twice,  or  .25. 

This  process  you  may  observe  conforms  to  the  rule,  to 
add  as  many  ciphers  as  may  be  necessary  to  render  the 
division  complete. 

3.  Reduce  the  vulgar  fraction  f  to  a  decimal.  30  is  com- 
posed of  the  prime  factors  3X2X5;  it  contains  2  only  once, 


ANALYSIS    OF   DECIMALS.  73 

and  therefore  it  is  not  divisible  by  2X2;  30  must  therefore 
be  multiplied  bj  10.  This  will  introduce  another  2,  and 
it  will  stand  thus,  2X2)3X2X5X:'X5.  By  expunging  the 
two  2's  and  multiplying  together  the  other  factors,  we  have 
.75  for  the  answer. 

4.  Reduce  the  fraction  ^^  to  a  decimal.  10  expressed  by  its 
factors  is  2X5,  and  8  is  2X2X2.  We. must  therefore  mul- 
tiply 2X5  by  10  till  it  shall  contain  the  factor  2  as  many 
times  as  8  contains  the  same  factor.  That  is,  the  numerator 
1  must  be  multiplied  by  a  thousand.  It  will  then  stand, 
2X2X2)  2X5X2X5X2X5.  Expunging  the  three  twos 
there  remains  for  the  answer  .125. 

By  examining  the  above  examples  you  may  observS  this 
fact,  that  if  the  denominator  of  the  vulgar  fraction  contains 
one  of  the  factors  of  10,  that  is,  2  or  5,  one  or  more  times  as 
a  factor,  the  decimal  will  contain  the  other  factor,  just  as 
many  times.  Thus,  ^=.5.  ^  or  ■i^^^=.25,  or  .5  X.5  ;  ^  or 
7x^xif^-^^^i  ^^  '^X'5X'5.  In  the  same  way  i=-2  ;  ^V  ^^ 
^^^=.04,  or  .2X.2;  ^^  or  ^x|^^=.008,  or  .2X.2X.2. 
In  this  way  you  may  determine  that  y^^  ^^^"  reduced  to  a 
decimal,  will  contain  5  four  times  as  a  factor,  because  16  con- 
tains 2  four  times  as  a  factor.  So  ^V  will  contain  5  five  times 
as  a  factor. 

This  is  conveniently  expressed  by  saying,  whatever  power 
of  one  of  the  factors  of  10  the  denominator  of  the  vulgar 
fraction  contains,  the  same  power  of  the  other  factor  will 
appear  in  the  decimal. 

5.  Reduce  ^  to  ^  decimal  fraction.  Preparing  the  num- 
bers as  before,  it  will  stand  3)2X5.  You  observe  that  3  is 
different  from  either  of  the  factors  of  10.  Now  as  10  has 
only  the  factors  2  and  5,  it  is  not  divisible  by  3  without  a 
remainder. 

If  you  add  to  the  numerator  ever  so  many  ciphers,  you  will 
only  increase  the  number  of  times  that  2  and  5  appear  in  it 
as  its  factors,  and  the  number  can  never  become  divisible  by 
3  without  a  remainder.  The  answer  becomes  .333-f-  and 
this  indefinitely,  as  far  as  you  may  please  to  carry  on  the 
operation.  On  the  same  principle  we  shall  find  that  it  is  not 
possible  to  express  accurately  in  decimals  any  vulgar  fraction 
whose  denominator  contains  as    a  factor   anything  different 


74  MENTAL    ARITHMETIC. 

from  the  factors  of  10  ;  for  this  denominator  becomes,  in  the 
reduction,  a  divisor  of  10  or  some  power  of  10,  and  if  it  has 
anything  in  it  as  a  factf  r  which  is  prime  to  the  factors  of  10, 
the  complete  division  is  impossible.  Thus  ^  cannot  be  ex- 
actly expressed  in  decimals ;  because,  though  one  of  its  fac- 
tors, 2,  is  a  divisor  of  10,  the  other,  3,  is  prime  to  10.  On 
this  principle  the  following  questions  .may  be  examined. 

Can  ^  be  accurately  expressed  in  decimals  ?     Why  ? 

Can  ^  be  accurately  expressed  in  decimals  ?     Why  ? 

Can  I  be  accurately  expressed  in  decimals  ?     Why  ? 

Can  -^^  be  accurately  expressed  in  decimals  ?     Why  ? 

Can^?  tV?  a?  a?  iiV?  rV?  iV?  ^?  T>s?  A?-Ar? 
A?  A?  iV?  ,     ^ 

6.  Name  all  the  denominators,  from  2  up  to  20,  of  such 
fractions  as  can  be  accurately  expressed  in  decimals  ? 

From  20  to  40  ?     From  40  to  60  ?     From  60  to  80  ? 

7.  Name  all  the  denominators,  from  2  to  20,  of  such  frac- 
tions as  cannot  be  expressed  accurately  in  decimals?  From 
20  to  40  ?     From  40  to  60  ?     From  60  to  80  ? 

8.  What  is  the  value  of  4  shillings  expressed  in  the  deci- 
mal of  a  £  ?  As  1  shilling  is  ^  of  a  £,  4  s.  is  -^.  We  can 
change  4  to  tenths  by  adding  a  cipher ;  it  will  then  be  40 ; 
4,  however,  was  not  the  number  we  wished  to  reduce  to  tenths, 
but  /^;  the  answer,  40,  is  therefore  20  times  too  great ;  dividing 
by  20  it  stands  .2.     4  shillings,  then,  is  2  tenths  of  a  £. 

9.  Now  reverse  the  operation  ;  what  is  the  value  in  shil- 
lings of  .2  of  a  £  ?  Now  shillings  are  twentieths ;  we  can 
change  any  number  to  twentieths  by  multiplying  it  by  20,  as 
1  is  20  twentieths,  2  is  40  twentieths,  &c.  Multiplying  the 
.2  by  20  we  have  40  ;  but  observe  the  two  was  not  two  wholes, 
but  two  tenths ;  the  answer,  40,  therefore,  is  ten  times  too 
great ;  dividing  by  10  the  answer  is  4  shillings. 

10.  Reduce  to  the  decimal  of  a  £,  2  shillings.     5  shillings. 

11.  What  is  the  value  in  shillings  of  .1  of  a  £  ?  of  .25  of 
a£? 

12.  Reduce  to  a  decimal  of  a  shilling,  3  pence.  3  pence 
are  y\  of  a  shilling ;  reducing  to  hundredths  to  render  the 
division  complete,  the  ans.  is  .25. 

13.  What  is  the  value  in  pence  of  .25  of  a  shilling  ? 

14.  Reduce  9  pence  to  the  decimal  of  a  shilling. 

15.  Reduce  1  peck  to  the  decimal  of  a  bushel. 

16.  Reduce  3  pecks  to  the  decimal  of  a  bushel. 

17.  Reduce  .5  of  a  bushel  to  pecks.    .75  of  a  bu.  to  pecks. 


INTEREST.  75 

18.  Reduce  15  minutes  to  the  decimal  of  an  hour. 

1 9.  Reduce  45  minutes  to  the  decimal  of  an  hour. 

20.  Reduce  to  minutes  .5  of  an  hour.     .25  of  an  hour.    .75 
of  un  hour. 

21.  Reduce  6  in.  to  the  dec.  of  a  foot.     9  in.  to  dec.  of  a 
ft.     3  in.  to  the  dec.  of  a  ft. 


SECTION    XIV. 

INTEREST. 

Interest  is.  the  sum  piiid  by  the  borrower  to  the  lender  for 
the  use  of  money.  The  rate  of  interest  is  established  by  law, 
and  varies  in  different  countries.  In  England  it  is  5  per 
cent.,  that  is,  5  for  the  use  of  100  for  1  year ;  in  the  New 
England  States  it  is  6  per  cent. ;  in  New  York  it  is  7  per 
cent.  When  no  particular  rate  is  mentioned  in  this  book,  6 
per  cent,  will  be  understood. 

If  I  borrow  100  dollars  fof  1  year,  at  the  end  of  the  year 
I  owe  the  sum  I  borrowed,  100  dollars,  and  6  dollars  for  the 
use  of  it,  making  106  dollars.  The  sum  borrowed  is  the  prin- 
cipal ;  the  sum  paid  for  the  use  of  it  is  the  interest ;  the  prin- 
cipal and  interest  added  together  make  the  amount. 

1.  What  is  the  interest  of  100  dolls,  for  2  years?  3  years  ? 
4  years  ?  5  years  ?  6  years  ?  7  years  ? 

2.  What  is  the  interest  of  200  dolls,  for  2  years  ?  3  years  ? 
4  years  ?  5  years  ?  6  years  ? 

3.  What  is  the  interest  of  300  dolls,  for  2  years  ?  for  4 
years  ?  of  400  dolls,  for  3  years  ? 

4.  What  is  the  interest  of  50  dolls,  for  1  year  ?  for  3  years  ? 
of  25  dolls,  for  1  year  ?  2  years  ? 

5.  What  is  the  interest  of  100  dolls,  for  1  year  ? 
What  is  the  interest  of  100  cents  for  1  year? 

What  is  the  interest  of  2  dolls,  for  1  year  ?  of  3  dolls.  ?  of 
4  dolls.?  5  dolls.?  6  dolls.?  7  dolls.?  8  dolls.?  9  dolls.? 

6.  What  is  the  interest  of  36  dolls,  for  1  year  ?  of  47  dolls.  ? 
of  57  dolls.  ?  of  34  dolls.  ?  of  62  dolls.  ?  of  89  dolls.  ?  of  125 
dolls.  ?  of  136  dolls.  ?  of  207  dolls.  ?  of  561  doUs.  ? 

7.  What  is  the  interest  of  50  cents  for  1  year  ?  of  25  cents  ? 


76  MENTAL   ARITHMETIC. 

of  10  cents  ?   of  20  cents  ?   of  30  cents  ?  of  40  cents  ?  of  50 
cents  ?  of  70  cents  ?  6f  80  cents  ?  of  90  cents  ? 

8.  What  is  the  interest  of  50  doll.  60  cents  for  1  year  ?  of 
84.30?  of  96.40?  of  112.25?  of  230.75? 

9.  What  is  the  interest  of  100  dolls,  for  6  months  ?  for  3 
months  ?  for  2  months  ?  for  1  month  ?  for  4  months  ?  for  5 
months  ?  for  7  months  ?  for  8  months  ?  for  9  months  ?  for  10 
months  ?  for  11  months  ? 

10.  What  is  the  interest  of  10  dolls,  for  6  mo.  ?  3  mo.  ?  2 
mo.  ?  1  mo.  ?  4  mo.  ?  5  mo.  ?  7  mo.  ?  8  mo.  ?  9  mo.  ?  10  mo.  ? 
'11  mo.? 

11.  What  is  the  interest  of  1  doll,  for  6  mo.  ?  1  mo.  ? 
The  interest  of  1  dollar  for  1  month  is  half  a  cent,  and  for 

any  number  of  months,  it  is  half  as  many  cents. 

12.  What  is  the  interest  of  1  dollar  for  5  mo.?  7  mo.?  8 
mo.?  9mo. ?  llmo. ?  12  mo.  ?  15mo. ?  16mo. ?  17mo. ? 
18  mo.  ? 

The  interest  of  any  number  of  dollars  for  1  month  is  half 
as  many  cents. 

13.  What  is  the  interest  of  12  dollars  for  1  mo.?  of  15 
dolls.  ?  25  doUs.?  37  dolls.?  42  dolls.?  67  dolls.?  93  dolls.? 
104  dolls.? 

14.  What  is  the  interest  of  12  dolls,  for  3  months? 
What  is  the  interest  of  25  dolls,  for  6  months  % 

In  computing  interest  a  month  is  reckoned  30  days.  As 
the  interest  on  a  dollar  for  30  days  is  half  a  cent,  that  is  5 
mills,  the  interest  on  a  dollar  for  1  fifth  of  30  days  will  be  1 
mill.  One  fifth  of  30  is  6  ;  the  interest  therefore  on  1  dollar 
for  6  days  is  1  mill,  and  the  interest  on  any  number  of  dollars 
for  6  days  will  be  as  many  mills  as  there  are  dollars. 

15.  What  is  the  interest  of  15  dollars  for  6  days  ?  of  25 
dolls.  ?  of  40  dolls.  ?  of  65  dolls.  ?  of  75  dolls.  ?  of  100  dolls.  ? 
of  500  dolls.  ?  of  360  dolls.  ?  of  840  dolls.  ?  of  1000  dolls.  ? 

As  the  interest  of  1  doll,  for  6  days  is  1  mill,  for  12  days 
it  will  be  2  mills,  for  18  days  3  mills,  &c. 

16.  What  is  the  interest  of  1  doll,  for  24  days  ?  of  2  dolls, 
for  6  days  ?  of  2  dolls,  for  12  days?  of  2  dolls,  for  18  days? 
of  5  dolls,  for  6  days?  for  12  days?  for  24  days?  of  36  dolls. 
for  18  days? 

17.  What  is  the  interest  of  125  dolls,  for  1  year  and  6  mo.  ? 


IKTEREST.  77 

18.  What  is  the  int.  of  268  dolls,  for  3  years  4  mo.  ? 

19.  What  is  the  int.  of  45  dolls,  for  4  years  7  mo.  ? 

20.  What  is  the  int.  of  60  dolls,  for  1  year  3  mo.  18  days  ? 
,21.  What  is  the  int.  of  100  dolls,  for  2  years  1  mo.  12  days? 

22.  What  is  the  int.  of  165  dolls,  for  3  years  2  mo.  6  days? 

23.  What  is  the  int.  of  50.45  for  1  year  7  mo.  12  days? 

24.  What  is  the  int.  of  94  dolls,  for  8  mo.  24  days  ? 

25.  What  is  the  int.  of  132.25  for  6  mo.  3  days? 

26.  What  is  the  int.  of  81.20  for  4  months  15  days  ? 

27.  What  is  the  int.  of  64.50  for  10  months  16  days? 

28.  What  is  the  int.  of  86  dolls,  for  9  days  ? 

29.  What  is  the  int.  of  340  dolls,  for  15  days  ? 

30.  What  is  the  int.  of  875  dolls,  for  22  days  ? 

When  interest  is  more  or  less  than  6  per  cent.,  first  find 
the  interest  at  6  per  cent,  and  then  make  a  proportional  addi- 
tion or  subtraction  for  the  required  per  cent.  If  it  is  7  per 
cent,  add  one  sixth ;.  if  5  per  cent,  subtract  one  sixth. 

31.  What  is  the  int.  of  140  dolls,  for  1  year, at 7  percent.? 

32.  What  is  the  int.  of  200  dolls,  for  1  year  and  6  mo.  at 
5  per  cent. 

33.  What  is  the  int.  of  460  dolls,  for  1  year  at  4^  per  cent. 

Remark.  —  4|^  is  three  fourths  of  6. 

34.  What  is  the  int.  of  500  dolls,  for  1  mo.  at  9  per  cent.  ? 


BANKING. 

When  money  is  obtained  at  a  Bank,  the  note  which  is  given 
for  it  promises  to  pay  it  at  a  certain  time,  as  60,  90,  or  120 
days.  The  interest  on  this  note,  instead  of  being  paid  at  the 
end  of  the  time,  when  the  note  is  taken  up,  is  paid  before 
hand ;  that  is,  it  is  subtracted  from  the  sum  named  in  the 
note ;  so  that,  when  you  take  up  the  note,  you  have  only  to 
pay  the  face  of  it,  as  the  interest  has  been  paid  already. 

If  you  give  a  note  to  a  Bank  for  100  dolls,  to  be  paid  in 
90  days,  they  subtract  from  the  sum  named  in  the  note  the 
interest  of  the  sum  for  90  days,  and  three  days  besides,  called 
days  of  grace  ;  the  balance  is  the  sum  you  receive.  The  in- 
terest of  100  dolls,  for  90  days  is  $1.50;  for  3  days  it  is 
7* 


78  MENTAL   ARITHMETIC. 

5  cents  ;  $1.55  subtracted  from  $100.00  leaves  a  balance  of 
$98.45,  which  is  the  sum  you  will  receive. 

If  the  note  is  given  for  60  days,  the  interest  is  cast  for  63 
days,  and  subtracted  from  the  sum  named. 

The  interest  thus  subtracted  is  called  the  bank  discount ; 
and  the  bank,  when  it  lends  money  on  such  a  note,  is  said  to 
discount  the  note. 

35.  What  is  the  bank  discount  on  a  note  of  100  dollars 
payable  in  30  days  ?  and  how  much  will  be  received  on  such 
a  note  ? 

The  interest  on  100  dollars  for  30  days  is  50  cents  ;  for  3 
days  it  is  5  cents ;  the  discount,  55  cents,  subtracted  from  100 
dollars,  leaves  $99.45,  the  sum  received. 

36.  What  is  the  bank  discount  on  a  note  for  200  dollars 
for  60  days  ?  and  what  is  the  cash  value  of  the  note  ? 

37.  What  is  the  bank  discount,  and  what  is  the  cash  value 
of  a  note  for  150  dollars  payable  in  30  days  ? 

38.  What  is  the  bank  discount,  and  what  is  the  cash  value 
of  a  note  for  200  dollars  payable  in  90  days  ? 

39.  What  is  the  bank  discount,  and  what  is  the  cash  value 
of  a  note  for  300  dollars  payable  in  90  days  ? 


DISCOUNT. 

When  money  is  paid  by  the  debtor  before  it  becomes  due, 
an  allowance  is  made,  which  is  called  discount.  If  I  owe  100 
dollars,  to  be  paid  in  three  months  from  this  time,  and  I  pay 
it  now,  I  ought  not  to  pay  the  full  hundred  dollars,  for  I  am 
entitled  to  the  use  of  the  money  three  months  longer.  The 
sum  which  should  be  paid  now,  to  cancel  a  debt  due  at  some 
future  time,  is  called  the  present  worth  of  the  debt. 

To  find  the  present  worthy  of  a  debt  due  at  some  future 
time,  first  find  the  interest  on  the  debt  from  the  time  of  pay- 
ment to  the  time  when  the  debt  is  .due  ;  subtract  this  interest 
from  the  debt,  and  the  remainder  will  be  the  present  worth. 
Thus,  if  I  pay  a  debt  of  100  dollars  three  months  before  it 
is  due,  I  subtract  the  interest  of  100  dollars  for  three  months, 
=  $1.50,  — from  100  dollars,  leaving  $98.50  for  the  sum 
which  I  must  pay. 

This  rule  is  not  strictly  equitable,  because  %  98.50,  with 
three  months'  interest  added,  will  not  amount  to  $100.  The 
above  method,  therefore,  gives  the  present  worth  a  little  too 


LOSS    AND    GAIN.  79 

small ;  but  it  is  the  method  uniformly  adopted  in  business, 
and  the  error  is  on  the  right  side,  for  it  encourages  the  debtor 
to  be  prompt  in  his  payments. 

40.  What  is  the  present  worth  of  200  dollars  payable  in 

1  year  ? 

41.  What  is  the  present  worth  of  150  dollars  payable  in 

2  years  ? 

42.  What  is  the  present  worth  of  60  dollars  payable  in 
6  months  ?  _ 

43.  What  is  the  present  worth  of  530  dollars  payable  in 
1  year? 

What  is  the  present  worth  of  400  dollars  payable  in  1  year 
and  6  months  ? 


LOSS  AND  GAIN.  — PER  CENTAGE. 

44.  A  boy  bought  a  penknife  for  25  cents,  and  sold  it  for 
28  cents ;  how  many  cents  did  he  gain  on  quarter  of  a  dollar  ? 

45.  Suppose  he  had  bought  4  knives  at  the  same  price 
each,  and  sold  them  at  the  same  profit,  he  would  then  have 
traded  with  a  dollar ;  how  much  would  he  have  gained  on  a 
dollar  ? 

This  is  called  so  much  per  cent,  which  only  means  so  much 
on  a  hundred. 

46.  A  boy  bought  a  bushel  of  apples  for  50  cents,  and  sold 
them  for  59  cents  ;  how  much  did  he  gain  per  cent.  ? 

47.  A  bookseller  bought  a  book  for  75  cents,  and  sold  it 
for  84  cents  ;  how  much  did  he  make  per  cent.? 

As  75  is  |-  of  100,  what  he  gained  on  the  book  will  be  |- 
of  what  he  would  gain  on  a  hundred ;  or  what  he  would  gain 
per  cent. 

48.  A  boy  bought  some  melons  for  40  cents,  and  sold  them 
for  60  cents  ;  wliat  did  he  make  per  cent.  ? 

Ans.  His  gain  was  equal  to  half  his  outlay. 

49.  A  grocer  bought  a  lot  of  flour  for  5  dollars  a  barrel, 
but  finding  it  damaged,  he  sold  it  for  4  dollars  a  barrel ;  what 
did  he  lose  per  cent.  ? 

50.  A  man  bought  a  share  in  a  bank  for  80  dollars,  and 
Bold  it  for  82  dollars  ;  what  did  he  gain  per  cent.  ? 

51.  A  man  bought  a  lot  of  apples  for  $1.50  a  barrel;  what 
must  he  sell  them  for  to  gain  10  per  cent.  ? 


80  MENTAL    ARITHMETIC. 

52.  A  hatter  bought  some  hats  for  $3.50  each ;  he  is  willing 
to  sell  them  at  a  profit  of  4  per  cent. ;  at  what  price  will  he 
sell  them  ? 

53.  A  manufacturing  company  declare  a  dividend  of  7^ 
per  cent. ;  what  ought  a  stockholder  to  receive  who  owns  350 
dollars  in  that  factory  ? 

54.  A  has  a  note  against  B  for  140  dollars,  which  he  sells 
for  cash  at  4  per  cent,  discount ;  what  does  he  receive  for  the 
note  ? 

55.  A  merchant  buys  100  barrels  of  flour  for  5  dollars  a 
barrel,  and  sells  it  so  as  to  lose  5  per  cent. ;  what  does  he  sell 
it  for  a  barrel  ? 

He  afterwards  buys  250  casks  of  lime  at  1  dollar  a  cask ;  he 

wishes  to  sell  it  so  as  to  make  good  his  loss  on  the  flour ;  at 

what  per  cent,  profit  must  he  sell  it,  and  for  how  much  a  cask  ? 

^You  observe  that  the  money  invested  in  lime  is  only  one 

half  as  much  as  was  invested  in  flour. 

56.  A  lends  B  10  dollars  for  2  months  without  interest ; 
afterwards  B  lends  /  5  dollars ;  how  long  can  A  keep  it  to 
balance  the  favor  he  iid  to  B  ? 

57.  C  lends  D  ICO  dollars  without  interest  for  4  months ; 
afterwards  D  lends  C  25  dollars  ;  how  long  can  C  keep  it  to 
balance  the  favor  ? 

In  these  cases  you  will  see  that  the  money  multiplied  by 
the  time  it  was  kept  must,  in  the  two  cases,  be  equal.  If  10 
dollars  is  lent  me  by  A  without  interest  for  6  months,  I  can 
balance  the  favor  by  lending  A  5  dollars  for  12  months,  or  4 
dollars  for  15  months,  or  15  dollars  for  4  months,  or  30  dollars 
for  2  months,  or  2  dollars  for  30  months,  or  20  dollars  for  3 
months. 

58.  A  lends  B  60  dollars  for  three  months,  without  requir- 
ing interest ;  afterwards  B  lends  A  90  dollars  ;  how  long  may 
A  keep  the  money  to  balance  the  favor  ? 

59.  A  lends  B  40  dollars  for  three  months  ;  afterwards  B 
lends  to  A,  for  two  months,  a  certain  sum,  the  use  of  wliich 
should  balance  the  favor  ;  how  large  must  the  sum  be  ? 

60.  A  lends  B  150  dollars  for  4  months;  B  afterwards 
lends  A  100  dollars ;  how  long  can  A  keep  it,  to  balance  the 
favor  ? 


SQUARE   MEASURE.  81, 

SECTION    XV. 

.SQUARE  MEASURE, 

Linear  measure  is  measure  in  a  straight  line,  having  length 
onlj.  Square  measure  is  the  measure  of  surface,  having 
length  and  breadth. 

Thus  a  linear  inch,     ■ — — 


A  square  inch, 


*A  line  of  2  inches,  when  square,  will  therefore  make  4 
square  inches,  thus, 


1.  A  line  of  three  inches,  when  square,  will  make  how 
many  square  inches  ? 

2.  The  square  of  4  inches  is  how  many  square  inches  ? 
The  square  of  0  inches  ?  of  6?  of  7  ?  of  8  ?  of  9  ?  of  10? 
of  11?   of  12? 

3.  How  many  square  inches  are  there  in  a  square  foot  ? 

*  Note  6. 


82 


MENTAL    ARITHMETIC. 


4.  How  many  linear  feet  are  there  in  a  linear  yard  ?  How 
many  square'feet  in  a  square  yard  ? 

5.  How  many  square  inches  are  there  in  a  piece  of  board 
"12  inches  long  and  3  inches  wide  ? 

6.  How  many  square  inches  are  there  in  a  piece  of  board  8 
inches  long  and  6  inches  wide  ?  in  a  board  5  inches  long  and 
3  inches  wide  ?  in  a  board  9  inches  long  and  5  inches  wide  ? 

7.  How  many  square  feet  are  there  in  the  floor  of  a  room 
12  feet  long  and  10  feet  wide  ? 

8.  How  many  square  feet  are  there  in  the  floor  of  an  entry 
15  feet  long  and  4  feet  wide  ? 

9.  How  many  square  yards  of  carpeting  will  cover  a  room 
6  yards  long  and  5  yards  wide  ? 

How  many  square  rods  are  there  in  a  piece  of  land  14  rods 
long  and  8  rods  wide  ? 

10.  If  a  road  4  rods  wide  passes  through  my  land  for  the 
distance  of  60  rods,  how  many  square  rods  of  my  land  does  it 


occupy .'' 

We  will  now  return  to  the  measure  of  an  inch, 
of  squaring   a  linear   inch,   we   take 
only  half  an  inch  and  square  it,  we 
shall  hav©^  but  one  fourth  of  a  square 
inch,  thus, 


If,  instead 


So  if  we  square  one  third  of  an  inch, 
it  will  give  us  ^  of  a  square  inch.  If 
we  square  one  fourth  of  an  inch,  it 
willgive  us  y*^  of  a  square  inch. 

11.  What  part  of  a  square  inch  will  -J-  of  an  inch  when 
squared  be  ?  What  part  of  a  square  inch  will  ^  of  an  inch 
be  when  squared  ?  ^?  ^?  ^?  -j^^?  ■^?  ■^^? 

12.  What  part  of  a  square  inch  is  a  piece  of  paper  1  inch 
long  and  half  an  inch  wide  ?  One  inch  long  and  three  fourths 
of  an  inch  wide  ? 

13.  What  part  of  a  square  foot  is  a  board  1  foot  long  and 
half  a  foot  wide  ?     One  foot  long  and  9  inches  wide  ? 

14.  How  many  square  inches  will  there  be  in  the  square 
of  a  line  1^  inches  long  ? 

[This,  and  the  following  questions  may  be  answered  by 
drawing  the  figure  on  a  slate  or  on  a  board.] 

How  many  square  inches  will  there  be  in  the  square  of  a 
line  2^  inches  long?  8^?  4^?  5^?  6^?  7^?  8^?  9^?  10|? 


SQUARE   MEASURE.  88 

15.  How  many  square  feet  in  ^  sl  yard  squared  ? 

16.  How  many  square  incbes  are  there  in  1  square  foot? 

17.  How  many  square  feet  in  1  square  yard  ? 

18.  How  many  square  yards  in  1  square  rod  ? 

19.  How  many  square  feet  in  1  square  rod? 

40  square,  rods  make  1  rood ;  4  roods  make  1  acre. 

20.  How  many  rods  make'  1  acre  ? 

21.  If  a  piece  of  board  is  6  inches  wide,  how  long  must  it 
be  to  contain  a  square  foot  ? 

22.  If  a  piece  of  board  is  3  inches  wide,  how  long  must  it 
be  to  contain  a  square  foot  ? 

23.  How  long  must  it  be  to  contain  a  square  foot,  if  it  is  2 
inches  wide  ?     If  1  inch  wide  ?     If  4  inches  wide  ? 

■■    24.  If  cloth  is  1^  a  yard  wide,  how  much  in  length  will 
make  a  square  yard  ? 

25.  How  much  lining  f  of  a  yard  wide  will  line  one  yard 
of  cloth  one  yard  wide  ? 

26.  If  cloth  is  two  thirds  of  a  yard  wide,  how  much  in 
length  will  it  take  for  a  square  yard  ? 

■  27.  How  much  cloth  ^  a  yard  wide  will  it  take  to  line  7 
yards  of  cloth  |  of  a  yard  wide  ? 

How  much  f  wide  will  line  l^-  yards  f  wide  ? 

28.  How  long  must  a  strip  of  land  one  rod  wide  be  to  con- 
tain an  acre  ?  How  long,  if  2  rods  wide  ?  If  3  rods  wide  ? 
If  4  rods  wide  ?     If  8  rods  wide  ?     If  10  rods  wide  ? 

29.  How  long  must  a  piece  of  land  be  to  contain  |  of  an 
acre,  if  it  is  4  rods  wide  ? 

30.  If  a  piece  of  land  is  10  rods  in  length  how  wide  must  it 
be  to  contain  ^  an  acre  ? 

31.  A  man  has  an  acre  of  land  16  rods  in  length ;  how  wide 
is  it? 

32.  How  many  steps  must  the  owner  take  to  walk  round  it 
if  he  take  5  steps  to  a  rod  ? 

33.  A  man  has  an  acre  of  land  8  rods  wide ;  how  long  is  it  ? 
How  many  rods  of  fence  will  it  take  to  fence  it  ? 

34.  If  a  road  4  rods  wide  is  laid  out  through  ray  land,  how 
much  of  the  road  will  it  take  in  length  to  make  an  acre  ? 
How  many  acres  will  there  be  -in  one  mile  of  the  road  ? 

35.  If  it  passes  through  my  land  for  half  a  mile,  and  I  am 
paid  at  the  rate  of  30  dollars  an  acre  for  the  land  occupied  by 
the  road,  what  will  be  the  amount  of  damages-  due  me  ? 


84  MENTAL   ARITHMETIC. 

36.  If  land  in  the  city- is  worth  45  cents  a  square  foot,  what 
will  be  the  cost  of  a  building-lot  30  feet  front  and  60  feet  from 
front  to  rear  ? 

37.  There  are  two  pieces  of  land  ;  one  of  them  12  rods 
square,  the  other  13.     Which  is  nearest  an  acre  ? 

38.  There  is  a  piece  of  land  12^  rods  square ;  how  much 
does  it  fall  short  of  an  acre  ? 

39.  A  painter  tells  me  it  will  cost  20  cents  a  square  yard 
to  paint  the  floor  of  a  room  in  my  house  ;  supposing  the  room 
is  5  yards  wide  and  6^  yards  long,  what  will  the  painting  of 
it  come  to  ? 

40.  What  will  the  painting  of  an  entry  cost,  at  the  same 
rate ;  that  is  1^  yards  wide  and  7  yards  in  length  ? 

41.  A  stone-cutter  agrees  to  lay  a  hammered  stone  door-step 
for  50  cents  for  every  square  foot  of  hammered  surface.  The 
stone  is  5  feet  long,  3^  feet  wide,  and  9  inches  thick ;  what  is 
the  surface  of  the  top,  the  two  ends,  and  the  front  edge,  added 
together  ?     What  will  be  the  cost  of  the  stone  ? 

42.  How  many  men  could  stand  on  |-  of  a  mile  square, 
allowing  each  man  1  square  yard  to  stand  upon  ? 

There  are  various  ways  of  finding  the  answer  to  the  above 
questions.  To  encourage  the  student's  invention,  some  of 
them  will  be  here 'suggested. 

First  Method.  As  there  are  dO^  square  yards  in  one  square 
rod,  multiply  80  (the  number  of  rods  in  one  fourth  of  a  mile,) 
by  itself;  and  this  product  by  30^.  80X80=6400  ;  6400  X 
30^=180,000-fl2,000+1600=:193,600,  answer. 

Second  Method.  Multiply  80  by  5J-,  which  will  give  the 
number  of  men  in  one  line  one  fourth  of  a  mile  long  ;  multiply 
this  product  by  itself.  80X5^=440;  4402=160,000+32,000 
+1600=193,600,  answer. 

There  are  still  other  ways  of  solving  the  question,  which 
the  student  may  discover  for  himself. 

Note.  For  those  students  who  have  not  time  to  go  through 
the  hook,  the  course  of  instruction  in  Mental  Arithmetic  may 
properly  close  at  this  place.  With  a  similar  view,  the  Second 
Part  may  be  divided  at  the  close  of  Section  XXXVI.  The 
ground  thus  gone  over  will  be  found  to  embrace  all  the  princi- 
ples and  practice  needed  in  the  transactions  of  ordinary 
business.  Those  whose  opportunities  permit,  should  have  the 
advantage  of  the  higher  discipline  furnished  in  the  remainder 
of  the  book. 


CONSTRUCTION   OF   THE   SQUARE.  85 

SECTION    XVI. 

CONSTRUCTION  OF  THE  SQUARE. 

If  I  place  three  dots  in  a  row,  and  place  three  such  rows 
side  by  side,  this  .will  represent  to  the  eye  the  square  of  the 
number  3. 

^  In  the  same  way  you  may  represent 

Thus,     •    •    *      the  square  of  4,  5,  or  any  number  what- 

9    #    S      ever.     I  will  now  ask  your  attention  to 

the  square  of  4.     We  may  make  it  by 

making  a  row  of  4  dots,  and  placing  4 

such  rows  side  by  side.     But  there  is  another  way  of  coming 

at  the  square  of  4.     We  will  take  the  square  of  3,  as  shown 

above,  and  see  what  additions  we  must  make  to  it,  in  order  to 

make  it  the  square  of  4*    You  observe  that  it  must  be  wider  by 

one  row  and  longer  by  one  row  than  it  is  now.    We  will  then 

add  a  row  above  the  others,  and  also  a  row  on  the  right  hand. 

I  have  made  the  additions  by  stars 

Thus,     •    #    •    >fc     to   distinguish   them   from  the   dots. 

You    now    see    there    is    something 

wanting  to  complete  the  square,  —  a 

sinojle  star  in  the  corner. 


* 

*  * 

• 

•  • 

* 

• 

•  • 

* 

• 

•  • 

* 

* 

*  * 

* 

• 

•  • 

* 

e 

•  • 

* 

• 

•  • 

* 

You  observe,  therefore,  that  you 
obtain  the  square  of  4  by  adding  to 
Thus,  ^  "^  ^  -^  the  square  of  3,  twice  3  plus  1.  We 
will  now  take  the  square  of  4,  and  by 
additions  to  it  obtain  the  square  of  5. 
Adding  a  row  of  4  at  the  top,  and  a 
row  of  4  at  the  right  hand,  there  will  be  one  \yanting  at  the 
corner  to  complete  the  square.  Adding  this,  which  makes 
twice  4-f-l,  we  have  the  square,  complete.  If  therefore  we 
have  the  square  of  any  number,  we  can  find  the  square  of  a 
number  one  greater  by  adding  twice  the  first  number  plus  1. 

The  square  of  5  is  25 ;  what  must  you  add  to  this  square  to 
make  the  square  of  6  ? 

What  must  you  add  to  the  square  of  6  to  make  the  square 
of  7  ?  What  must  you  add  to  the  square  of  7  to  make  the 
square  of  8  ? 

S 


86,  MENTAL    ARITHMETIC. 

What  must  you  add  to  the  square  of  9  to  make  the  squaro 
of  10? 

The  square  of  15  is  225  ;  what  is  the  square  of  16,  by  the 
above  method  ? 

The  square  of  20  is  400  ;  what  is  the  square  of  21  ? 

The  square  of  30  is  900  ;  what  is  the  square  of  31  ? 

The  square  of  40  is  1600  ;  what  is  the  square  of  41  ? 

The  square  of  50  is  2500  ;  what  is  the  square  of  51  ? 

What  is  the  square  of  60  ?  of  61  ?  of  70  ?  of  71  ?  of  80  ? 
of  81  ?   of  90  ?  of  91  ? 

We  will  now  return  to  the  square  of  3,  and  I  ask  your  close 
attention  once  more.  Supposing  we  have  the  square  of  3 
l^efore  us,  and  we  wish  to  make  such  additions  to  it  as  shall 
make  the  square  of  5.  As  5  is  two  greater  than  3,  we  must 
add  two  rows  instead  of  one.  If  we  add  2  rows  of  3  at  the 
top,  and  2  rows  of  3  at  the  right  hand,  the  figure  will  stand 
thus,    _    _    _    .  .    .  . 

Here  you  see  there  are  four  stars 
wanting  to  complete  the  square.  I 
have  marked  their  places  by  the 
circle  (O).  If  you  suppose  these 
four  to  be  added  the  square  will  be 
complete,  and  will  be  the  square  of 
5.  The  question  is  now,  what  has 
been  added  to  the  square  of  3  in  order  to  make  the  square  of 
5  ?  You  observe  there  are  added  6  stars  or  two  rows  of 
three  at  the  top,  6  on  the  right  hand,  and  4  in  the  corner,  to 
make  the  square  of  5.  But  we  can  express  this  in  a  different 
way.  We  may  consider  5  as  consisting  of  two  parts,  3  and 
2  added  together.  We  will  call  3  the  first  part,  and  2  the 
second  part  of  5.  Now  by  the  figure  you  perceive  that  the 
square  of  5  is  made  up,  first,  of  the  square  of  the  first  part, 
that  is,  the  nine  dots ;  then  the  stars  at  the  top  are  the  product 
of  the  first  part  multiplied  by  the  second,  and  adding  to  these 
the  stars  on  the  right  hand,  we  have  twice  the  product  of  the 
first  part  into  the  second ;  and,  last,  we  have,  in  the  corner, 
the  square  of  the  second  part. 

To  state  it  briefly  once  more  :  regarding  5  as  made  up  of 
the  two  parts,  3  and  2,  the  square  of  5  we  find  is  equal  to  the 
square  of  the  first  part-f-twice  the  product  of  the  two  parts+ 
the  square  of  the  second  part. 


* 

* 

* 

O  O 

* 

* 

* 

o  o 

• 

• 

• 

*  * 

• 

9 

• 

*  * 

• 

• 

• 

*  * 

THE    SQUARE    OP   FRACTIONAL   NUMBERS.  87 

This  is  called  expressing  the  amount  of  a  square  in  the 
terms  of  its  parts. 

Examine  and  answer  the  following  questions  : 

1.  If  we  regard  the  number  6  as  made  up  of  two  parts,  4 
and  2,  how  will  you  express  the  square  of  6  in  the  terms  of 
its  parts  ? 

2.  Regard  the  number  7  as  consisting  of  two  parts,  5  and 
2 ;  what  is  the  square  of  7  in  the  terms  of  its  parts  ? 

You  can  draw  the  figure  for  yourself  and  see  the  applica- 
tion of  the  principle  in  the  above  cases. 

It  is  of  no  consequence  in  what  way  the  number  is  divided ; 
the  operation  will  bring  out  the  exact  square  of  the  whole 
number  in  all  cases.  To  show  this  we  will  take  the  number 
10,  the  square  of  which  is  100.  We  will  first  divide  10  into 
the  parts,  7  and  3  ;  then,  by  the  formula  given  above,  the  sq. 
of  7-|-twice  the  product  of  7  into  3-f-  sq.  of  3,  will  be  the  sq. 
of  the  whole  number.  The  sq.  of  7=49,  twice  7  X  3=42,  the 
sq.  of  3=9  ;  49+42+9=100. 

We  will  now  divide  10  into  the  parts  6  and  4,  proceeding  as 
above  ;  we  find  36+48+16=100. 

Again,  we  will  divide  10  into  the  equal  parts,  5  and  5, 
25+50+25=100.  * 

Finally  divide  10  into  the  parts  8  and  2.     64+32+4=100. 

We  will  now  apply  the  above  method  to  the  purpose  of 
finding  some  squares  of  larger  numbers. 

3.  What  is  the  sq.  of  25  ?  dividing  into  20+5  ;  Ans.  400+ 
200+25=625  ? 

4.  What  is  the  sq.  of  35,  or  30+5  ?  Ans.  900+300+ 
25=1225. 

5.  What  is  the  sq.  of- 46  or  40+6?  Ans.  1600+480+ 
36=2116. 

6.  What  is  the  sq.  of  55  ?  of  64?  of  75  ?  of  83  ?  of  92  ? 

7.  What  is  the  sq.  of  125  ?  divide  into  100+25.  100  sq. 
=10000,  twice  100X25=5000,  25  sq.=625.     Ans.  15,625. 

8.  What  is  the  square  of  150  ?  of  230  ?  of  510. 

The  same  formula  will  embrace  the  examples  mentioned  in 
the  first  part  of  this  section,  when  the  second  part  of  the  num- 
ber is  1,  for  example, 

9.  What  is  the  sq.  of  5  or  4+1  ?  Here  twice  the  product 
of  the  two  parts  is  merely  twice  the  first  part,  inasmuch  as 
multiplying  by  1  does  not  increase  the  number  ;  and  the  sq. 
of  1  is  only  1.  The  answer,  therefore,  by  the  formula,  is  16 
+8+1=25. 


88.  '  MENTAL    ARITHMETIC. 

10.  This  metliod  will  apply  to  the  squaring  a  whole  num- 
ber and  a  fraction,  as  follows :  What  is  the  sq.  of  1+^.  Ans. 
1+1-|-^=2;|,  for  twice  the  product  of  ^  into  1  is  1,  and  the 
sq.  of  1^  is  :^. 

To  test  the  correctness  of  this  answer,  we  will  perform  the 
operation  another  way,  1^=|,  |  sq.=|=2:^. 

11.  Whatisthesq.  of  21-?  3^?  4^?  5^?  6^?  7^? 

The  answer  in  each  of  these  cases  may  be  tested  by  chang- 
ing the  mixed  number  to  an  improper  fraction,  as  2|=f,  &c. 
We  may  in  the  same  way  square  the  sum  of  two  fractions. 

12.  What  is  the  sq.  of  ^+^?     Ans.  i+i+i=l. 
Now  i-|-i=l,  the  sq.  of  which  is  1. 

13.  What  is  the  sq.  of  f+J  ?  Ans.  tV+i^+TV=i|-  ov  1. 
Now  |-(-i=l,  the  sq.  of  which  is  1. 

14.  What  is  the  sq.  of  f+J?  The  sum  of  these  is  2,  the 
sq.  of  which  is  4 ;  the  answer,  therefore,  should  be  4.  Apply- 
ing the  formula,  the  operation  is  as  follows ;  1+1+1=  ^=4. 

PE ACTIO AL  QUESTIONS. 

We  will  now  apply  the  above  principle  to  the  solution  of 
some  questions  which  appear  at  first  a  little  difficult. 

1.  A  boy  had  some  apples ;  he  placed  part  of  them  in  rows 
making  a  square,  and  found  he  had  6  apples  left.  He  placed 
another  row  on  two  sides,  and  found  he  had  enough  to  com- 
plete the  square  except  one  apple  at  the  corner.  How  many 
apples  were  there  in  the  first  square  ?  How  many  apples 
had  he  ? 

2.  Three  boys  were  playing  at  marbles ;  the  first  says,  I 
have  just  marbles  enough  to  make  a  square ;  and  he  placed 
them  in  rows  on  the  floor,  forming  a  square ;  the  second  boy 
says,  I  have  1 2  marbles,  and  I  will  put  a  row  on  two  sides  of 
yours,  and  make  your  square  larger ;  but  on  placing  his  mar- 
bles he  found  he  wanted  3  more  to  complete  the  square ;  then 
the  third  boy  says,  I  have  just  three,  and  that  will  make  the 
square  complete. 

How  many  had  the  first  boy  ?  How  large  was  the  square 
which  all  the  marbles  made  ? 

3.  The  boys  of  a  school  thought,  one  day,  at  recess,  they 
would  form  themselves  into  a  square.  A  part  of  them  first 
formed  a  square,  when  itrwas  found  that  there  were  15  boys 
left ;  these  15  then  placed  themselves  in  a  row  on  two  sides  of 


COMPLETING    OF    THE    DEFECTIVE    SQUARE.  80- 

the  square,  when  it  was  found  that  it  required  2  boys  more  to 
complete  the  square.  How  many  boys  were  there  in  the  first 
square  ?     How  many  in  all  ? 

4.  A  general,  drawing  up  his  soldiers  in  a  square  body,  with 
the  same  number  in  rank  and  file,  found  he  had  55  men  over 
and  above.  He  placed  these  in  a  row  on  two  sides,  and  found 
that  he  now  wanted  30  men  to  complete  the  square.  How 
many  men  were  there  on  a  side  of  the  first  square  ?  How 
many  men  in  the  first  square  ?     How  many  men  had  he  in  all  ? 

5.  There  is  a  certain  square  number  expressed  in  the  terms 
of  its  parts,  that  is,  it  is  expressed  in  three  terms,  the  first  of 
which  is  the  square  of  the  first  part,  the  second  is  twice  the 
product  of  the  two  parts,  and  the  third  is  the  square  of  the 
second  part.  Now  the  first  two  terms  are  16-|-24,  what  is 
the  third  term  ?     What  is  the  number  ? 

6.  There  is  a  square  number  expressed  in  the  terms  of  its 
parts ;  the  first  two  terms  are  9-|-24  ;  what  is  the  third  ? 

7.  The  first  and  last  terms  of  a  square  are  4-(-  25  ;  what^ 
must  be  the  middle  term  ?     What  is  the  number  ? 

8.  The  first  and  last  terms  of  a  sq.  are  9+4 ;  what  is  the 
second  term  ?     What  is  the  square  ? 

9.  Complete  the  square  whose  first  two  terms  are  16-|-40 
+  D. 

10.  Complete  the  sq.  36+24+ Q. 

11.  Complete  the  sq.  4+4+ D . 

12.  Complete  the  sq.  9+6+ □. 

13.  Complete  the  sq.  16+8+IZ]. 

14.  Complete  the  sq.  25+10+a. 

15.  Complete  the  sq.  25+20+CI3. 

16.  Complete  the  sq.  36+72+a. 

17.  Complete  the  sq.  25+30+a. 

18.  Complete  the  sq.  25+40+a. 

The  square  root  is  the  number  which  multiplied  into  itself 
produces  the  square.  Thus  3  is  the  sq.  root  of  9,  2  is  the  sq. 
root  of  4,  5  is  the  sq.  root  of  25. 

The  square  root  of  a  square  of  three  terms,  like  those 
given  above,  is  the  sq.  root  of  the  first  term,  plus  the  square 
root  of  the  third,  for  these  multiplied  by  themselves  will  pro- 
duce the  square.  Thus  the  square  root  of  the  square  9+12 
+4  is  3+2,  or  5.  3  is  the  sq.  root  of  9,  and  2  the  sq.  root  of 
4.  This  number,  3+2,  multiplied  by  itself,  will  produce  the 
square  of  9+12+4. 
8* 


90  MENTAL   ARITHMETIC. 

19.  Complete  the  sq.  36+60+n.     What  is  its  root? 

20.  Complete  the  sq.  36-f-12-(-a.     What  is  its  root? 

21.  Complete  the  sq.  16-|-44-n.     What  is  its  root? 

22.  Complete  the  sq.  9+12=|-[Z3.     What  is  its  root? 

23.  What  is  the  square  root  of  169  ? 

Divide  the  number  into  three  terms,  100+60-|-9.  We 
divide  it  so  because  100  is  a  square,  and  60  is  twice  the  pro- 
duct of  10,  the  root  of  the  first  term,  into  the  root  of  what  this 
way  of  dividing  loaves  us  for  the  third  term.  That  is,  if  we 
take  60  for  the  2d  term,  we  leave  9  for  the  3d  term,  and  this 
is  as  it  should  be,  for  60  is  twice  the  product  of  10  into  3. 

The  root  is  .therefore  10+3,  or  13. 

24.  What  is  the  sq.  root  of  196  ? 

We  will  take  for  the  first  term  100,  whose  root  is  10.  Now 
as  the  2d  term  is  twice  the  product  of  the  two  terms  of  the 
root,  if  we  divide  half  of  it  by  the  1st,  it  will  give  the  2d  term 
of  the  root ;  or  what  is  the  same  thing,  if  we  divide  the  2d 
term  of  the  square  by  twice  the  first  term  of  the  root,  it 
will  give  the  2d  of  the  root.  Now  96  contains  the  2d  and 
third  terms  of  the  square  ;  we  must  separate  it  into  two  parts, 
such  that  the  first  part,  divided  by  twice  10,  or  20,  will  give 
for  quotient  the  root  of  the  second  part. 

Let  us  first  try  by  dividing  it  into  60  and  36.  Now  60 
divided  by  20  gives  three,  which  is  not  the  root  of  36;  our 
division,  therefore,  was  wrong.  The  2d  term  was  too  small, 
and  the  3d  too  great.  We  will  try  again.  By  dividing  it 
into  80  and  16,  we  find  that  80,  divided  by  20,  gives  4,  which 
is  the  root  of  16.  The  number  196,  when  arranged  in  the 
three  terms  of  the  square,  will  be  lOO-j-80-1-16,  and  the  root 
is  10+4,  or  14. 

25.  What  is  the  root  of  225  ?  Here  we  must  not  take  for 
our  first  term  200,  for  this  is  not  a  square.  We  must  take  the 
largest  square  whose  root  is  an  even  10.  This  is  100.  We 
have  remaining  125  ;  this  we  must  divide  into  two  terms, 
such  that  the  first  divided  by  twice  10  will  give  the  root  of  the 
second  term.  We  will  first  divide  into  80  and  45  ;  80  di- 
vided by  20  gives  4,  which  is  not  the  square  root  of  45.  We 
will  divide  into  100  and  .25  ;  100  divided  by  20  gives  5, 
which  is  the  exact  root  of  25.  The  number  225,  therefore, 
when  arranged  in  the  three  terms  of  a  square,  stands  100+ 
100+25,  and  its  square  root  is  10+5,  or  15. 

26.  What  is  the  square  root  of  256  ?     Taking  for  the  first 


MENSURATION.  91 

term  100,  it  remains  to  divide  the  remainder,  156,  according 
to  the  principle  stated  above.  Now  120  will  contain  20,  6 
times,  which  is  the  root  of  the  remainder,  36 ;  the  number 
stands,  therefore,  100-|-120-(-36 ;  square  root  lO-f-6,  or  16. 

27V  What  is  the  square  root  of  484  ?  Here  we  take  for  the 
first  tel-m  400,  for  that  is  the  largest  square  whose  root  is  in 
even  tens ;  its  root  is  20.  The  remainder  we  may  divide  into 
80  and  4  ;  dividing  80  by  twice  20,  or  40,  we  have  for  the  quo- 
tient 2,  which  is  the  root  of  the  third  term.  The  square, 
therefore,  is  400-f-80+4.     The  root,  20+2,  or  22. 

28.  Whatisthesquarerootof 529?  of576?  of  625?  of  676? 

29.  The  following  is  a  ready  method  of  squaring  a  mixed 
number  whose  fraction  is  ^ ;  as  2^,  3|,  4^.  The  fraction  will 
be  ^  ;  for  the  other  number,  multiply  the  whole  number  by  a 
number  greater  than  itself  by  one.  Thus  the  square  of  2^; 
the  fraction  is  ^^  the  whole  number  2X3  or  6,  6^.  The 
square  of  3^,  3X4,  or  12,  and  ^.  The  sq.  of  4^  is  4X5,  or  20, 
and  I.     What  is  the  sq.  of  5^?  6^?  7^?  8|?  9^?  lOi 

The  same  principle  will  apply  to  the  square  of  whole  num- 
bers whose  last  figure  is  5 ;  as  25,  45,  55,  &c.,  for  such  a 
number  consists  of  a  certain  number  of  tens,  and  half  of  10. 
As  the  right  hand  figure  is  5,  the  two  right  hand  figures  of 
the  square  must  be  25  ;  then  multiply  the  number  at  the  left 
of  the  5  by  itself  increased  by  1,  and  this,  read  at  the  left 
hand  of  the  25,  will  be  the  square.  Thus,  for  the  sq.  of  25, 
the  two  right  hand  figures  will  be  25  ;  for  the  rest,  multiply  2 
by  3,  which  is  6 ;  ans.  625. 

30.  What  is  the  sq.  of  35?  3X4=12;  to  this  annex  25; 
1225,  ans. 

What  is  the  sq.  of  45  ?  of  55  ?  of  65  ?  of  75  ?  of  85  ?  of  95  ? 


SECTION    XVII. 

PRACTICAL  QUESTIONS  IN  SQUAEE   MEASURE. 

1.  How  many  square  rods  are  there  in  a  square  mile  ? 

2.  How  many  acres  are  there  in  a  square  mile  ? 

3.  Divide  a  square  mile  into  4  equal  farms  ;  how  many  acres 
would  there  be  in  each  ? 

4.  How  many  acres  are  there  in  1  fourth  of  a  mile  square  ? 

5.  How  many  acres  are  there  in  a  town  6  miles  long  and 
5  miles  broad  ? 


^2  MENTAL    ARITHMETIC. 

6.  If  half  of  the  town  is  unfit  for  improvement  in  conse- 
quence of  water  and  mountains,  how  many  farms  of  100  acres 
might  be  made  from  the  other  half? 

7.  A  man  bought  a  rectangular  piece  of  land  containing  40- 
acres  ;  on  going  out  with  his  son  to  measure  round  it,  to  as- 
certain how  much  fence  it  would  require  to  enclose  it,  they 
found  the  first  side  they  measured  to  be  160  rods. 

We  need  not  measure  any  more,  said  the  son,  for  I  can  tell 
all  the  rest  in  my  head. 

How  wide  was  the  piece  ?  and  how  many  rods  of  fence 
would  it  take  to  go  round  it  ? 

8.  A  man  bought  7  acres  of  land,  in  rectangular  form ;  the 
width  of  it  was  28  rods  ?  what  was  its  length  ? 

If  a  four  sided  piece  of  land  is  rectangular,  its  contents  may 
be  found  by  multiplying  two  adjacent  sides,  or  sides  that  meet 
and  form  a  corner. 


12  c 

Thus,    '         ~ 


4      if  a  piece  is  12  rods  long 


and  4  rods  wide,  the  two  boundaries,  12  and  4,  which  meet 
and  form  the  right  angle  at  c,  will,  when  multiplied  together, 
give  the  contents  in  square  rods  ;  12X4=48. 

If  the  opposite  sides  are  parallel,  but  the  angles  are  not 
right  angles,  the  distance  between  the  two  sides  must  be 
measured  by  a  perpendicular  line,  thus  : 

— y    the  length,  16  rods,  multiplied 

/      by  the  width,  4  rods,  will  give 
the  contents. 


If  the  piece  is  a  triangle,  there  must  be  a  perpendicular 
Tine  drawn  to  the  longest  side  from  the  angle  opposite  to  it. 
This  perpendicular  we  may  call  the  height  of  the  triangle, 
and  the  longest  side,  its  length;  and  the  height  multiplied 
into  the  length  will  give  double  the  area ;  dividing  this  by  2 
we  get  the  area. 

To  show  the  reason  of  this,  take  the  following  figure :  by 

16       examining  this    you   will   see 

that  there  are  in  it  two  trian- 
gles just  alike  ;  the  length  of 
each  is  1 6  rods,  and  the  height 
4  rods.    Now  16X4:  will  give,  as  in  the  case  above,  the  area  of 


MENSURATION. 


93 


the  whole  figure ;  that  is,  of  both  the  triangles ;  therefore  it 
will  give  twice  the  area  of  one  of  them. 

9.  What  is  the  area  of  a  triangle  whose  longest  side  is  16 
rods,  and  the  perpendicular,  from  the  opposite  angle  to  this 
side,  12  rods  ? 

10.  What  is  the  area  of  a  triangle  whose  longest  side  is  24 
rods,  and  its  height  9  rods  ? 

11.  What  is  the  area  of  a  triangle  whose  longest  side  is  18 
rods,  and  height  1 6  rods  ? 

12.  A  triangle  contains  just  one  acre :  its  longest  side  is  20 
rods  ;  how  long  must  the  perpendicular  be  from  the  opposite 
angle  to  that  side  ?  ■^ 

13.  A  triangle  contains  2  acres ;  its  longest  side  is  32  rods ; 
how  long  is  the  perpendicular,  from  the  opposite  angle  to  this 
side? 

In  a  right  angled  triangle  the  longest  side  is  called  the 
hypotenuse ;  the  sides  containing  the  right  angle  are  called 
the  legs,  or  one  the  base,  and  the  other  the  perpendicular.  In 
all  right  angled  triangles,  the  square  of  the  hypotenuse  is  just 
equal  to  the  sum  of  the  squares  of  the  two  other  sides.  This 
important  principle  is  exhibited  to  the  eye  in  the  following 


figure. 


The  hypotenuse  is 
divided  into  5  equal 
parts,  and  its  square 
is  therefore  25.  The 
base  has  4  equal  parts 
of  the  same  length, 
making  a  square  of 
1 6  ;  the  perpendicu- 
lar is  divided  into  3 
equal  parts,  of  the 
same  length  as  the 
others,  which  makes 
a  square  of  9.  The 
square  of  the  perpen- 
dicular and  of  the 
base  added  together, 
16+9=25,  which  is 
the  square  of  the  hy- 
potenuse. 

If  we  know  the  square  of  the  hypotenuse,  we  know  the  sum 
of  the  squares  of  the  two  legs.     If  we  know  the  sum  of  the 


94 


MENTAL    ARITHMETIC. 


squares  of  the  two  legs,  we  know  the  square  of  the  hypote- 
nuse. If  we  know  the  square  of  the  hypotenuse  and  of  one 
leg  we  can  find  the  square  of  the  other  leg.  And  if  we  know 
the  square  of  any  one  of  these  sides  we  can  obtain  the  length 
of  the  side  by  extracting  the  square  root. 

14.  In  a  certain  right  angled  triangle  the  square  of  the 
hypotenuse  is  100  feet;  what  is  the  length  of  the  hypotenuse  ? 
In  the  same  triangle  the  square  of  the  base  is  64  feet ;  what 
is  the  length  of  the  base  ? 

In  the  same  triangle,  what  must  be  the  square  of  the  per- 
pendicular ?     What  is  the  length  of  the  perpendicular  ? 

15.  A  and  B  set  out  from*  the  same  place  ;  A  travels  east 
6  miles,  B  travels  north  till  he  is  10  miles  in  a  straight  line 
from  A ;  how  far  north  has  B  traveled  ? 

16.  There  is  a  triangle,  the  perpendicular  of  which  is  3 
feet,  the  hypotenuse  is  5  feet ;  how  long  is  the  base  ? 

17.  A  man  had  a  piece  of  land  in  the  form  of  a  right- 
angled  triangle,  the  two  legs  of  which  were  equal  to  each 
other,  and  the  square  of  the  hypotenuse  was  128  rods;  how 
many  rods  were  there  in  the  piece  ? 


The  circumference  of  a  cir- 
cle is  3  times  and  ^  greater 
than  the  diameter.  If  the  di- 
ameter is  1  foot,  the  circum- 
ference will  be  34-  feet ;  if  the 
diameter  is  2  feet,  the  circum- 
ference will  be  6f  feet. 


18.  If  the  diameter  of  a  circle  is  3  feet,  what  will  be  the 
circumference  ?  If  the  diameter  is  4  feet  ?  If  5  feet  ?  If  6 
feet?     If  7  feet? 

If  the  diameter  of  a  water  wheel  is  16  feet,  what  is  the 
circumference  ? 

19.  If  the  diameter  of  the  earth  is  8,000  miles,  what  is 
the  circumference  ? 

To  find  the  area  of  a  sector  of  a  circle,  as  a,  e,  b,  multiply 
the  arc  by  half  the  radius.  This  figure  may  be  regarded  as 
a  triangle,  the  base  of  which  is  the  arc,  and  the  radius,  the 


ANALYSIS    OF   PROBLEMS.  95 

height ;  and  you  have  seen  before,  that  in  a  triangle,  the  base 
multiplied  by  half  the  height  gives  the  area.  From  this,  we 
may  obtain  a  method  of  obtaining  the  area  of  the  whole 
circle. 

Multiply  the  circumference  by  half  the  radius.  For  we 
may  regard  the  circle  as  made  up  of  a  great  number  of  small 
triangles,  whose  bases  added  together  are  the  circumference 
of  the  circle,  and  whose  height  is  equal  to  radius ;  being  in 
each  case  the  distance  from  the  circumference  to  the  center. 

20.  What  is  the  circumference  of  a  circle  whose  diameter 
is  14  feet? 

What  is  its  area  ? 

21.  What  is  the  circumference  of  a  circle  whose  diameter 
is  12  feet  ? 

What  is  its  area,  ? 

22.  What  is  the  circumference  of  a  circle  whose  diameter 
is  20  feet? 

What  is  its  area  ? 

23.  What  is  the  circumference  of  a  circle  whose  diameter 
is  28  feet? 

What  is  the  area  ? 


SECTION    XVIII. 

ANALYSIS    OF    PEOBLEMS. 

1.  A  boy  spent  one  half  the  money  he  had,  and  had  1  dol- 
lar left ;  how  much  had  he  at  first  ? 

2.  A  boy  spent  one  third  of  the  money  he  had,  and  had 
1  dollar  left ;  how  much  had  he  at  first  ? 

Ans.  If  he  lost  1  third,  he  had  2  thirds  left ;  if  1  dollar 
was  two  thirds,  half  a  dollar  must  be  one  third,  and  f  of  a 
dollar  the  whole.     He  had  1  dollar  and  a  half. 

3.  A  boy  spent  ^  of  his  money,  and  had  1  dollar  left ;  how 
much  had  he  at  first  ? 

Let  this  and  the  following  answers  be  given  in  form  of  a 
fraction,  like  the  preceding  answer. 

4.  A  boy  spent  ^  of  his  money,  and  had  1  dollar  left ;  how 
much  had  he  at  first  ? 


96  MENTAL    ARITHMETIC. 

5.  A  hoj  spent  J  of  his  money,  and  had  1  dollar  left ;  how 
much  had  he  at  first  ? 

6.  A  boy  spent  |  of  his  money,  and  had  1  dollar  left ;  how 
much  had  he  at  first  ? 

7.  A  boy  spent  ^  of  his  money,  and  had  1  dollar  left ;  how 
much  had  he  at  first  ? 

8.  A  boy  spent  ^  of  his  money,  and  had  1  dollar  left ;  how 
much  had  he  at  first  ? 

9.  A  boy  spent  -^  of  his  money,  and  had  1  dollar  left ; 
how  much  had  he  at  first? 

10.  A  man  carried  some  corn  to  mill;  the  miller  took  -^ 
of  it  for  toll,  and  then  there  was  just  a  bushel;  how  much 
did  the  man  carry  to  mill  ? 

11.  A  man  carried  some  cloth  to  be  fulled;  it  shrank  two 
sevenths  in  its  length,  and  was  then  just  a  yard  long;  how 
long  was  it  at  first  ? 

12.  A  man  drew  a  prize  in  a  lottery ;  ^  of  the  prize  was 
retained,  and  then  the  drawer  received  just  100  dollars ;  how 
much  was  the  prize  ? 

13.  If  a  stick  of  timber  shrink  ^  in  weight  in  seasoning, 
and  then  weigh  100  pounds,  how  much  did  it  weigh  at  first  ? 

14.  A  teamster  sold  f  of  a  cord  of  wood,  and  then  had 
half  a  cord  left ;  how  much  had  he  at  first  ? 

15.  A  man  had  an  estate  left  him  by  his  father;  he  lost  J 
of  it;  he  then  received  1000  dollars,  and  then  he  had  3500 
dollars ;  how  much  had  he  at  first  ? 

16.  A  merchant  began  trade  with  a  sum  of  money,  and 
gained  so  as  to  increase  his  original  stock  by  ^  of  itself;  he 
then  lost  500  dollars,  and  had  4,500  dollars  left ;  how  much 
did  he  begin  with  ? 

17.  A  man  set  out  on  a  journey,  and  spent  half  the  money 
he  had  for  a  dinner ;  he  then  paid  half  of  what  he  had  left  for 
provender  for  his  horse ;  then,  half  of  what  now  remained 
for  toll  in  crossing  a  bridge,  and  had  10  cents  left;  how  much 
had  he  at  first  ? 

18.  A  boy  spent  f  of  his  money  for  a  book,  and  i^  of  it  for 
some  paper,  and  had  8  cents  left ;  how  much  had  he  at  first  ? 

19.  A  boy  playing  at  marbles  lost,  in  the  first  game,  ^  of 
what  he  had ;  in  the  second  game,  I  of  what  he  then  had ; 
in  the  third,  ^  of  what  he  then  had;  in  the  fourth  11,  and 
then  he  had  16  marbles  left;  how  many  had  he  at  first ? 

20.  A  boy  playing  at  marbles,  wins  in  the  first  game  so  as 


ANALYSIS    OF   PROBLEMS.  97 

to  double  the  number  of  marbles  he  had ;  in  the  second  game, 
he  loses  ^  of  what  he  then  had ;  in  the  third  game,  he  loses 
5,  and  then  finds  he  has  just  as  many  as  at  first ;  how  many 
had  he  at  first  ? 

21.  A  man  had  his  sheep  in  three  pens ;  in  the  first  ther^ 
were  10  sheep,  in  the  second  there  were  as  many  as  in  the 
first,  and  half  the  number  in  the  third ;  in  the  third  there 
were  as  many  as  in  the  first  and  second;  how  many  had  he 
in  all  ?  ■ 

22.  In  an  orchard  ^  of  the  trees  are  plumb  trees ;  there 
are  20  cherry  trees ;  and  the  apple  trees,  which  constitute  the 
remainder,  are  half  as  many  as  the  plumb  and  cherry  trees 
added  together ;  how  many  trees  are  there  in  the  orchard  ? 

23.  John  and  William  were  talking  of  their  ages,  John 
says,  I  am  twielve  years  old ;  William  says,  if  half  my  age, 
were  multiplied  by  one  fourth  of  yours,  and  half  your  age  plus 
one  subtracted  from  the  product,  that  would  give  my  age. 
How  old  was  he  ? 

24.  A  man  talking  of  the  age  of  his  two  children,  said 
the  youngest  was  three  years  old ;  the  age  of  the  eldest  was 
^  his  own  age;  if  his  own  age  was  divided  by  that  of  his 
youngest,  and  once  and  one  third  the  age  of  the  youngest 
subtracted  from  the  quotient,  that  would  give  the  age  of  the 
eldest;  how  old  was  the  eldest? 

25.  The  number  of  pupils  in  a  school  is  such  that  if  you 
take  half  of  them,  and  increase  that  by  2 ;  then  take  one 
third  of  this  last  number ;  and  increase  it  by  3,  and  from  this 
number  subtract  6,  the  remainder  will  be  7 ;  how  many  are 
there  in  the  school  ? 

26.  A  boy  plays  three  games  at  marbles ;  in  the  first,  he 
loses  a  certain  number ;  in  the  second,  he  gains  8 ;  in  the 
third,  he  loses  4,  and  then  he  finds  he  has  two  more  than  he 
began  with  ;  how  many  did  he  lose  in  the  first  game  ? 

27.  A  boy  playing  at  marbles  first  lost  one  third  of  what 
he  had ;  he  then  doubled  his  number,  when  he  had  5  marbles 
more  than  he  had  at  first ;  how  many  had  he  at  first  ? 

28.  There  is  a  certain  number,  one  third  of  which  exceeds 
one  fourth  of  it  by  2 ;  what  is  the  number  ? 

29.  There  is  a  certain  number,  one  fourth  of  which  exceeds 
one  fifth  of  it  by  1 ;  what  is  the  number  ? 

30.  There  is  a  certain  number,  one  third  of  which  added 
to  one  fifth  of  it  amounts  to  16 ;  what  is  the  number? 

9 


98  MENTAL   ARITHMETIC. 

31.  There  is  a  number,  one  third,  one  fourth,  and  one  fifth 
of  which  added  together  are  94;  what  is  the  number? 

32.  What  is  that  number  a  fifth  of  which  exceeds  a  sixth 
of  it  by  4? 

33.  What  number  is  that  of  which  a  fourth  part  exceeds 
a  seventh  part  by  9  ? 

34.  In  a  certain  orchard  there  are  apple,  peach,  and  pear 
trees  ;  the  apple  trees  are  two  more  than  half  the  whole ;  the 
peach  trees  are  one  third  of  the  whole,  and  are  14  less  than 
the  apple  trees  ;  the  rest  are  pear  trees  ;  how  many  are  there 
of  each  kind,  and  how  many  in  all  ? 


SECTION    XIX. 

SOLID  MEASURE. 

Whatever  has  length  and  breadth,  and  thickness  is  a  solid. 
A  block  of  wood  1  inch  long,  1  inch  high,  and  1  inch  wide, 
is  a  solid  inch.  A  block  1  foot  long,  1  foot  wide,  and  1  foot 
high,  is  a  solid  foot.  A  block  1  yard  long,  1  yard  wide,  1 
yard  high,  is  a  solid  yard. 

1.  How  many  solid  inches  are  there  in  a  block  3  in.  long,  2 
in.  wide,  and  1  in.  high  ? 

2.  How  many  in  a  block  4  in.  long,  3  in.  wide,  and  2  in. 
high? 

3.  How  many  solid  feet  are  there  in  a  block  5  feet  long,  3 
feet  wide,  and  2  feet  high  ? 

4.  How  many  solid  feet  in  a  block  7  jTeet  long,  2  feet  wide, 
and  2  feet  high  ? 

When  a  solid  has  its  length,  height,  and  breadth,  equal  to 
each  other,  it  is  called  a  cube  ;  and  the  linear  measure  of  its 
length,  height,  or  breadth  is  called  the  root  of  the  cube.  We 
have  seen  what  is  a  cubic  inch,  a  cubic  foot,  and  a  cubic  yard. 

Suppose  now  we  have  a  pile  of  cubic  inch  blocks,  and  we 
wish  to  construct  from  them  a  cube,  each  of  whose  dimensions 
shall  be  2  inches ;  we  will  first  take  2  blocks,  and  place  them 
down  side  by  side ;  this  will  be  as  long  as  the  required  figure, 
but  it  will  not  be  wide  enough,  nor  high  enough  j  to  make  it 


SOLID   MEASURE.  99 

wide  enough,  we  will  place  2  more  blocks-down  by  the  side 
of  the  former ;  the  figure  now  contains  4  cubic  inches,  and  is 
2  inches  long,  and  2  inches  wide,  but  it  is  only  1  inch  high. 
To  make  it  2  inches-  high,  we  must  place  upon  this  another 
layer  of  4  blocks  arranged  just  like  the  former.  The  figure 
will  then  be  2  in.  long,  2  in.  wide,  and  2  in.  high ;  it  contains 
8  cubic  inches,  and  is  the  cube  of  2. 

5.  How  many  blocks  will  you  require,  and  how  will  you 
arrange  them  to  make  the  cube  of  3  ?  '9 

6.  How  many  blocks  will  you  re(|uire,  and  how  will  you 
arrange  them  to  make  the  cube  of  4  ? 

7.  How  many  blocks  will  you  require,  and  how  will  you 
arrange  them  to  make  the  cube  of  5  ? 

The  cube  when  expressed  in  numbers  is  the  same  as  the 
3d  power  of  the  root.  It  is  found  by  taking  the  root  3  times 
as  a  factor.  Thus  the  3d  power  of  2  is  2X2X2=8.  The 
3d  power  of3  is  3X3X3=27.  Of  4,  is  4X4X4=64.  Of 
5,  is  5X5X5=125. 

In  this  way  we  may  find  the  3d  power  of  any  number. 

8.  How  many  blocks,  of  a  cubic  foot  each,  will  it  take  to 
form  a  cubic  solid,  6  ft.  on  a  side  ? 

9.  How  many  blocks  of  a  cubic  foot  each  will  it  take  to  form 
a  cubic  solid  of  7  feet  each  way  ? 

10.  How  many  cubic  feet  will  it  take  to  form  a  cube  of  8 
feet? 

11.  How  many  cubic  feet  will  it  take  to  form  a  cube  of  9 
feet? 

12.  How  many  cubic  feet  will  it  take  to  form  a  cube  of  10 
feet? 

13.  How  many  cubic  inches  are  there  in  a  cubic  foot  ? 

•    14.  A  pile  of  wood  8  feet  long,  4  feet  high,  and  4  feet  wide, 
makes  a  cord  ;  how  many  cubic  feet  are  there  in  a  cord  ? 

15.  A  pile  of  wood,  4  feet  loog,  4  feet  high,  and  1  foot 
thick,  makes  what  is  called  a  cord  foot ;  how  many  cubic  feet 
are  there  in  a  cord  foot  ? 

16.  How  many  cord  feet  are  there  in  a  cord? 

17.  There  is  a  pile  of  wood  40  feet  long,  4  feet  wide,  and  5 
feet  high ;  how  many  cords  does  it  contain  ? 

18.  There  is  a  stick  of  hewn  timber  25  feet  long,  1  foot 
wide,  and  1  foot  thick  ;  how  many  cubic  feet  does  it  contain  ? 

19.  -There  is  a  tree  from  the  but-end  of  which  a  stick  may 
Se  hewn  13  feet  long,  2  feet  wide,  and  2  feet  thick ;  how  many 
•rnbic  feet  will  it  contain  ? 


100  MENTAL   ARITHMETIC. 

20.  It  is  estimated  that  50  feet  of  hewn  timber  weigh  a 
ton ;  if  50  cubic  feet  weigh  20  cwt.  net  weight,  what  will  1 
foot  weigh  ? 

21.  If  you  divide  a  cubic  inch  into  blocks  measuring  ^  an 
inch  each  way,  h.ow  many  such  will  there  be  in  a  cubic  inch  ? 

22.  How  many  cubic  half  inches  are  in  a  cubic  inch  ? 

23.  If  you  divide  a  cubic  inch  into  cubes  of  ^  of  an  inch 
each,  how  many  such  will  there  be  ? 

24.  How  many  cubic  quarter  inches  are  there  in  a  cubic 
inch? 

25.  How  many  cubic  inches  are  there  in  a  cube  of  one  inch 
and  a  half? 

26.  K  a  man  digs  a  cellar  at  the  rate  of  f  of  a  dollar  for 
a  cubic  yard,  what^ill  the  job  come  to,  if  the  cellar  is  18  feet 
long,  12  feet  wide,  and  6  feet  deep  ? 

27.  A  stone-layer  agreed  to  build  a  solid  wall  30  feet  long, 
4^  feet  thick,  and  6  feet  high,  for  2^  dollars  a  cubic  yard , 
what  did  the  wall  cost  ? 


CONSTRUCTION  OF  THE  CUBE. 

We  have  seen  that  the  third  power,  or  cube  of  any  number, 
is  obtained  by  taking  the  number  three  times  as  a  factor ;  the 
product  is  the  cube,  or  third  power. 

In  this  way  the  cube  of  any  number  whatever  may  be  ob- 
tained. There  is  another  way,  however,  of  constructing  the 
cube,  the  knowledge  of  which  is  very  important  in  the  ope- 
ration of  extracting  the  cube  root. 

Suppose  we  wish  to  find  the  cube  of  5.  Instead  of  taking 
5  three  times  as  a  factor,  thus,  5X5X5=125,  we  wiU  re- 
gard the  number  5  as  consisting  of  two  parts,  3  and  2.  We 
will  call  3  the  first  part,  and  2  the  second  part  of  5. 

We  will  begin  by  making  the  cube  of  the  first  part  3,  thus, 
3X3X3=27. 

We  will  regard  this  as  a  cube  of  3  inches,  that  is,  3  in.  long, 
3  in.  wide,  and  3  in.  high ;  and  represent  it  by  the  following 
figure. 

The  question  now  is,  how  shall  we  enlarge  this  cube  of  3,  so 
as  to  make  it  the  cube  of  5  ?  It  is  evident,  it  must  be  2  in. 
longer,  2  in.  broader,  and  2  in.  higher  than  it  now  is.     We 


CONSTRUCTION   O^  TEE   CUBE.  101 

y^Hf  begin*  <ten  by  putling  '2  layers 
of  inch  blocks  on  the  front  side,  2 
layers  on  the  right  side,  and  2  layers 
on  the  top.  The  figure  thus  en- 
larged is  not  a  cube.  There  are  sev- 
eral places  not  filled  up.  It  is  nearer 
the  cube  of  5  than  it  was  before; 
but  something  more  must  be  added. 
Before  making  that  addition,  how- 
ever, let  us  see  what  we  have  done.  The  figure  is  the  cube 
of  3,  which  is  the  first  part  of  5.  To  this  there  are  3  equal 
additions  made.  Each  of  these  additions  is  3  in.  square,  and 
2  in.  thick.  Now  3  is  the  first  part  of  5 ;  each  addition, 
therefore,  contains  the  square  of  the  first  part,  3,  multiplied 
by  the  second  part,  2,  or  3^X2;  therefore  the  three  addi- 
tions will  be  3  times  the  square  of  the  first  part  multiplied 
by  the  second. 

The  whole  figure,  therefore,  after  these  three  additions  are 
made,  contains  S^-j-S  times  3^X2. 

We  will  now  see  what  additions  must  next  be  made  to  the 
figure. 

There  are  3  places  that  need  filling  up,  each  3  in.  long, 
2  in.  wide,  and  2  in.  high.     Each  of  these  new  additions  is 

2  in.  square,  and  3^  in.  long.  It  consists,  therefore,  of  the  first 
part  of  5  multiplied  by  the  sq.  of  the  second ;  and  the  three 
together  are  3  times  the  first  part  multiplied  by  the  sq.  of 
the  second.  There  is  one  addition  wanting  to  complete  the 
cube  ;  that  is  at  the  corner.  It  must  be  2  in.  long  2  in. 
wide,  and  2  in.  high ;  that  is,  the  cube  of  2 ;  or,  in  other 
words,  the  cube  of  the  2d  part. 

Remembering  that  the  two  parts  of  5,  as  here  divided,  are 

3  and  2 ;  the  printed  figure  is  the  cube  of  the  first  part,  the 
first  addition  is  3  times  the  square  of  the  1st  part,  multiplied 
by  the  2d ;  the  2d  addition  is  3  times  the  first  part,  multiplied 
by  the  sq.  of  the  2d ;  the  3d  addition  is  the  cube  of  the  2d 
part.  Let  the  letter  a  stand  for  the  first  part,  3 ;  and  the 
letter  b,  for  the  2d  part,  2.  The  printed  figure  will  then  be  a^ ; 
the  first  addition,  3a2b ;  the  second  addition,  Sab^  ;  the  third 
addition,  b^.  The  whole  cube,  therefore,  will  be  a^-j-Sa^b-j- 
3ab2-|-b3.  Observe  that  the  letters  and  numbers  are  to  be 
multiplied  together  though  there  is  no  sign  of  multiplicatiou 

9* 


102  MENTAL  AKITaMETIC. 

betweenthem/^  da^b  i«*;t]liree.  iimes  the  square  of  a,  multi- 
plied by  b\  '        '-   . 

These  are  called  the  four  terms  of  the  cube,  when  the  root 
is  in  two  parts. 

If  we  express  the  above  in  the  numbers  for  the  cube  of  5, 
it  will  stand  thus 

-~r^  1st.  2d.  3d.  4th. 

y^y  33+3x32x2+3X3X22+2^. 

1.  What  number  makes  the  1st  term  of  this  cube? 

2.  What  number  forms  the  2d  term  ? 

3.  What  number  forms  the  3d  term  ? 

4.  What  number  forms  the  4th  term  ? 

5.  What  do  all  the  4  terms  amount  to  ? 

6.  Which  of  the  four  terms  contains  the  third  power  of  the 
first  part  ?     Which  contains  the  2d  power  of  the  first  part  ? 

7.  Which  contains  the  first  power  of  the  first  part  ?  Which 
term  contains  the  1st  power  of  the  second  part  ?  Which  the 
2d  power  ?     Which  the  3d  ? 

8.  If  the  4th  term  of  the  above  cube  were  not  given,  how 
could  you  determine  from  the  others  what  it  must  be  ? 

9.  If  the  3d  term  were  gone,  how  could  you  restore  it  ?  If 
the  2d  was  gone  how  could  you  restore  it  ? 

10.  If  you  divide  the  number  5  into  the  two  parts,  4  and  1, 
and  express  the  cube  according  to  the  above  rule,  what  will 
the  1st  term  be  ?  What  will  be  the  2d  term  ?  What  will  be 
the  3d  term?     What  the  4th? 

Remember  here,  that  all  powers  of  one  are  one,  —  neither 
more  nor  less. 

Divide  the  number  6  into  4+2,  and  form  the  cube  accord- 
ing to  the  above  rule. 

11.  What  will  the  1st  term  be?  The  2d?  The  3d? 
The  4th?      , 

12.  What  will  they  all  amount  to 

Multiply  6  into  itself  3  times,  thus,  6X^X6,  and  see  if  it 
amounts  to  the  same. 

13.  Divide  6  into  the  parts  5+1,  and  form  the  cube. 
What  will  be  the  1st  teym  ?  The  2d  ?  The  3d  ?  The  4th  ? 
What  do  they  all  amount  to  ? 

1 4.  There  is  a  cube  in  4  terms,  the  first  two  terms  of  which 
are  33+3X3^X1  ;  what  must  be  the  3d  term?  AYhat  the 
4th  term  ?    What  is  the  number  of  the  cube  ?    What  is  the 


THE    CUBE.  108 

root  of  the  cube  ?     This  root  is  the  cube  root  of  the  first 
term,  added  to  the  cube  root  of  the  last. 

15.  There  is  a  cube  in  4  terms,  the  first  two  of  which  are 
23-|-3X22X2;  what  is  the  third  term?  What  the  fourth? 
What  is  the  whole  cube  ?^  What  is  the  first  part  of  the  root  ? 
What  is  the  second  part  ?     What  is  the  whole  root  ? 

16.  Complete  the  cube  43-f3x42x2-H:ZI+23. 
What  is  the  number  of  the  cube  ?     What  the  root  ? 

17.  Complete  the  cube  33+3x32 x3+3X3x32-fa. 
What  is  the  number  of  the  cube  ? 

18.  There  is  a  cube  in  4  terms,. the  first  of  which  is  1000; 
what  is  the  first  part  of  the  root  ? 

19.  The  second  term  of  the  same  cube  is  300X6,  or  1800 ; 
what  is  the  third  term  ? 

20.  What  is  the  fourth  term  of  the  same^cube  ? 

21.  AVhat  is  ihe  root  of  the  above  cube  ? 

22.  The  first  term  of  a  cube  is  1000,  the  second  is  300X8, 
or  2400  ;  what  is  the  third  term  ? 

23.  What  is  the  fourth  terra  of  the  abov6  cube  ? 

24.  The  first  term  of  a  cube  is  8000,  the  second  is  1200X3 ; 
what  is  the  third  term  ? 

Observe  that  1200  is  3  times  the  square  of  the  first  term; 
consequently,  one  third  of  it  is  the  square  of  the  first  term. 

25.  What  is  the  fourth  term  in  the  above  cube  ?  What  is 
the  root  ? 


SECTION    XX. 

RATIO.  —PROPORTION. 

If  we  compare  the  two  numbers,  3  and  9,  in  order  to  ascer- 
tain their  relative  magnitude,  we  may  subtract  3  from  9  ;  we 
find  the  difference  to  be  6. 

There  is  another  way  of  comparing4he  two  numbers.  We 
may  see  how  many  times  3  will  go  in  9 ;  we  shall  find  the 
quotient  to  be  3. 

The  numbers  we  obtain  in  each  of  these  comparisons  is 
called  the  Ratio  of  the  two  numbers ;  but  they  differ  in  kind ; 


104  MENTAL    ARITBMETIC. 

the  former. is  called  Ai'ithmetical  ratio;  the  latter,  Geometri- 
cal ratio. 

Arithmetical  ratio,  then,  expresses  the  difference  of  two 
numbers ;  Geometrical  ratio  expresses  the  quotient  of  one 
of  the  numbers,  divided  by  the  otlier.  As  we  shall  speak 
only  of  geometrical  ratio  in  what  follows  here,  the  word 
ratio  whenever  it  is  used,  may  be  understood  to  mean  geo- 
metrical ratio.  The  ratio  of  4  to  2,  written  4  :  2,  is  2  ;  for  2 
will  go  in  4  twice.  The  ratio  of  12  to  3,  written  12  ;  3,  is  4; 
for  3  will  go  in  12,  4  times. 

The  two  numbers  compared,  are  together  called  the  terms 
of  the  ratio,  or  simply,  the  ratio  ;  the  first  is  called  the  Ante- 
cedent, the  second  is  called  the  Consequent.  These  two  terms, 
you  will  perceive,  correspond  exactly  to  the  numerator  and 
denominator  of  a  fraction ;  for  in  a  fraction,  the  numerator  is 
divided  by  the  denominator.  A  ratio  is  then  another  way  of 
expressing  a  fraction.  The  antecedent  is  the  numerator  ;  the 
consequent  the  denominator.  4  :  2  is  the  same  as  f ;  6  :  3,  the 
same  as  §. 

As  a  ratio  is  essentially  the  same  as  a  fraction,  everything 
is  true  of  a  ratio  which  is  true  of  a  fraction. 

1.  What  effect  will  it  have  on  the  value  of  the  ratio,  if 
you  increase  the  antecedent  ?  if  you  diminish  the  antecedent  ? 
if  you  double  the  antecedent  ?  if  you  divide  the  antecedent 
by  2? 

2.  What  effect  will  it  have  on  the  value  of  the  ratio,  if 
you  increase  the  consequent?  if  you  diminish  the  conse- 
quent ?  if  you  multiply  the  consequent  ?  if  you  divide  the 
consequent  ? 

3.  Take  the  ratio  4:2;  how  can  you  multiply  it  by  2  ?  In 
what  other  way  ? 

4.  How  can  you  divide  it  by  2  ?     In  what  other  way  ? 

5.  Take  the  ratio  6  ;  3  ;  how  can  you  multiply  it  by  2  ? 
Can  you  do  it  in  more  than  one  way  ?     If  you  cannot,  why  ? 

6.  How  can  you  divide  it  by  4  ?  Can  you  do  it  in  more 
than  one  way  ?     If  not,  why  ? 

Take  the  ratio  4:2;  multiply  both  terms  by  the  same  num- 
ber, 3,  for  example ;  it  will  be  12  :  6 ;  you  see  the  value  is  not 
altered ;  divide  both  terms  4:2  by  2  ;  it  will  be  2  :  1 ;  tho 
value  is  not  altered ;  it  is  still  2. 


PROPORTION.  105 


PROPORTION. 


If.  there  are  four  numbers,  and  the  first  has  the  same  ratio 
to  the  second  that  the  third  has  to  the  fourth,  the  four  num- 
bers are  said  to  be  in  proportion.  Thus  "the  numbers  2:1;: 
12:6,  are  in  proportion.  The  first  has  the  same  ratio  to  the 
second,  that  the  third  has  to  the  fourth.     The  ratio  is  2. 

A  Proportion  then  is  the  equality  of  two  Ratios. 

The  four  dots  : :  between  the  two  ratios,  are  the  same  as 
the  sign  of  equality,  =. 

In  order  to  preserve  the  proportion,  the  two  ratios  must  al- 
ways be  equal  to  each  other.  You  may  make  any  change 
you  please  in  the  terms,  provided  you  do  not  destroy  this 
equality. 

Let  us  take  the  proportion  4  :  2  : :  12  :  6 ;  the  value  of  the 
two  ratios  is  now  equal. 

1st.  Multiply  the  antecedents  by  2  ;  8  :  2  ::  24  :  6 ;  the 
numbers  are  still  in  proportion,  for  the  value  of  the  two  ratios 
is  equal. 

2d.  Divide  the  antecedents  by  2  ;  2  :  2  : :  6  :  6 ;  the  value 
of  the  two  ratios  is  equal. 

3d.  Multiply  the  consequents  by  2 ;  4  :  4  : :  12:12;  the 
value  of  the  two  ratios  is  equal. 

4th.  Divide  the  consequents  by2;4:l::12:3;  the  value 
of  the  ratios  is  still  equal. 

5th.  Multiply  both  terms  of  the  first  ratio  by  2 ;  8  :  4  : :  12  : 
6 ;  or  multiply  the  2  terms  of  the  second  ratio  by  2 ;  4:2:: 
24  :  12 ;  the  ratios  are  still  equal.  In  the  same  way  we  might 
take  any  other  number  for  our  operations  instead  of  2 ;  the 
same  operations  might  be  performed  without  destroying  the 
proportion. 

The  two  middle  terms  of  a  proportion  are  called  the  means ; 
the  first  and  last  terms  are  called  the  extremes. 

In  a  proportion  the  product  of  the  two  means  is  equal  to 
the  product  of  the  two  extremes. 

Take  the  proportion  4  :  2  : :  6  :  3  ;  the  product  of  the  means 
2X6  is  12  ;  and  the  product  of  the  extremes,  4X3,  is  12. 

Take  the  proportion  6  :  2  : :  9  :  3  ;  2X9=18,  3X6=18. 

Take  the  proportion  10 .:  2  : :  30  :  6  ;   2X30=6X10. 

If  we  know,  then,  the  product  of  the  means,  we  know  the 
product  of  the  extremes. 


106  MENTAL    ARITHMETIC. 

In  a  certain  proportion  the  product  of  the  means  is  30 ; 
what  must  be  the  product  of  the  extremes  ? 

6.  Further,  if  we  know  the  product  of  the  means,  and  if 
we  know  one  of  the  extremes,  we  can  find  the  other.  If,  as 
in  the  above  case,  the  product  of  the  means  is  30,  and  if  one 
of  the  extremes  is  3,  what  must  be  the  other  ? 

How  do  you  find  that  number  ? 

7.  If  the  product  of  the  means  is  30,  and  one  of  the  ex- 
tremes is  10,  what  must  the  other  be  ? 

How  do  you  find  the  number? 

8.  If  the  product  of  the  means  is  30,  and  one  of  the  ex- 
tremes is  5,  what  is  the  other  ? 

If  one  of  the  extremes  is  6,  what  is  the  other  ? 

If  one  of  the  extremes  is  15,  what  is  the  other? 

You  see,  therefore,  that  if  you  muUiply  the  means  togeth- 
er, and  divide  the  product  by  one  extreme,  the  quotient  will 
be  the  other  extreme. 

9.  If  the  product  of  the  means  is  72,  and  one  of  the  ex- 
tremes is  24,  what  will  the  other  be  ? 

10.  If  the  first  3  terms  of  a  proportion  are  9:6::  12, 
what  must  the  fourth  term  be  ? 

11.  "What  is  the  fourth  term  of  the  proportion  5  :  3  : :  15? 

12.  Complete  the  proportion  8  :  6  : :  12. 

13.  Complete  the  proportion  14  :  8  : :  7. 

14.  Complete  the  proportion  10  :  4  : :  15. 

By  means  of  this  rule,  many  interesting  questions  may  be 
solved. 

15.  If  8  yards  of  cloth  cost  6  dollars,  what  will  20  yards 
of  the  same  cloth  cost? 

It  is  evident,  that  the  length  of  the  shorter  piece  is  to  the 
length  of  the  longer,  as  the  cost  of  the  shorter,  'is  to  the  cost 
of  the  longer.  Now  we  know  all  these  numbers  except  the 
last,  and  can  express  them  in  a  form  of  a  proportion,  thus, 

Yd«.    Yds.       Dolls. 

8  :  20  : :  6.  8  is  the  length  of  the  shorter  piece ;  20,  the 
length  of  the  longer ;  6  is  the  cost  of  the  shorter  piece.  The 
fourth  term,  that  is,  the  cost  of  the  longer  piece,  we  have  not 
yet  found ;  you  must  discover  that  yourself;  how  can  you  do 
it? 

16.  If  18  yards  of  cloth  cost  15  dollars,  what  will  12  yards 
of  the  same  cloth  cost  ? 

What  do  you  here  seek,  —  the  quantity  of  clothe  or  tho 
price  ?    Is  it  the  price  of  the  longer,  or  of  the  shorter  piece  ? 


PROPORTION.  107 

How  can  you  make  a  proportion  with  the  two  quantities 
of  cloth,  and  the  two  sums  they  cost  ?  State  this  proportion 
in  general  terms,  putting  the  thing  sought  as  the  fourth  term. 
State  the  proportion  in  figures  all  except  the  fourth  term. 
How  will  you  find  the  'fourth  term  ? 

17.  If  5  yards  of  cloth  cost  2  dollars,  what  will  7  yards  of 
the  same  cloth  cost  ? 

State  the  proportion  in  general  terms. 

State  the  first  three  terms  in  figures,  and  find  the  fourth. 

18.  If  a  horse  travels  16  miles  in  3  hours,  how  far  will 
he  travel  in  2  hours  ? 

As  the  longer  time  is  to  the  shorter  time,  so  is  the  greater 
distance  to  the  smaller  distance. 

Remember  that  things  of  the  same  kind  should  stand  in 
the  same  ratio ;  and  that  the  quantity  sought  must  be  the 
fourth  term.  Then  inquire  what  the  true  proportion  must  be, 
and  state  it  in  general  terms,  repeating  the  trial,  if  neces- 
sary, till  you  perceive  that  you  are  right.  This  is  far  better 
than  any  special  rule,  for  it  leads  you  to  reason  on  what  you 
do. 

19.  If  a  certain  number  of  cubic  feet  of  timber  weighs  a 
certain  number  of  hundred  weight,  and  if  we  wish  to  know, 
without  weighing,  how  many  hundred  weight  a  certain  small- 
er number  of  cubic  feet  will  weigh,' what  will  be  the  propor- 
tion in  general  terms  ? 

20.  If  16  cubic  feet  of  wood  weigh  5  cwt,  what  will  6 
cubic  feet  weigh  ? 

21.  If  3  barrels  of  flour  last  a  family  7  months,  how  many 
barrels  will  last  them  12  months  ? 

22.  If  an  iron  rod  of  equal  size  throughout  and  of  a  certain 
length,  Aveighs  a  certain  number  of  pounds,  and  is  broken  into 
two  parts,  not  in  the  middle,  how  can  you  find  the  weight  of 
one  of  the  parts  without  weighing  it  ? 

23.  If  an  iron  rod  7  feet  long  weighs  20  pounds,  what  will 
5  feet  of  it  weisrh  ? 


COMPARISON  OF  SBHLAR  SURFACES. 

As  all  the  above  questions  may  be  answered  by  analysis  aa 
well  as  by  proportion,  the  rule  of  proportion  might  be  dis- 
pensed with  for  the  purposes  of  solving  this  kind  of  questions 


108  MENTAL   ARITHMETIC. 

It  has,  however,  very  important  and  interesting  applications 
in  the  measurement  of  similar  surfaces  and  solids.  To  pre- 
pare for  this,  you  must  attend  carefully  to  a  few  introductory 
statements. 

Two  surfaces  are  similar  to  each  other  when  they  are 
shaped  alike,  though  they  may  be  unequal  in  size.  Thus  a 
large  circle  is  similar  to  a  small  circle,  for  they  are  both 
shaped  alike. 

So  one  square  is  similar  to  another,  though  they  may  be 
unequal  in  size.  One  equilateral  triangle  is  similar  to  anoth- 
er equilateral  triangle. 

If  a  rectangle  is  twice  as  long  as  it  is  wide,  and  a  larger 
or  a  smaller  rectangle  is  twice  as  long  as  it  is  wide,  the  two 
are  similar.  In  the  same  way  any  two  surfaces,  however 
irregular  their  shape,  are  similar,  provided  they  are  shaped 
alike. . 

We  will  now  come  to  a  stricter  definition  of  similar  sur- 
faces. Similar  surfaces  are  such  as  have  their  corresponding 
dimensions  proportional.  Take  the  circle :  the  dimensions  of 
the  circle  are  the  diameter  and  the  circumference ;  the  "diam- 
eter of  one  circle  is  to  its  circumference,  as  the  diameter  of  a 
larger  or  a  smaller  circle  is  to  its  circumference.  For  you 
have  learned  before  that  the  circumference  of  a  circle  is  3^ 
times  its  diameter. 

Take  the  square ;  one  side  of  a  square  is  to  another  side 
of  it,  as  one  side  of  a  larger  or  smaller  square  is  to  another 
side  of  it. 

If  a  rectangle  is  twice  as  long  as  it  is  wide,  another  rec- 
tangle in  order  to  be  similar,  whatever  be  its  size,  must  be 
twice  as  long  as  it  is  wide. 

1 .  There  are  two  similar  rectangles  ;  one  is  8  feet  long 
and  6  feet  wide  ;  the  other  is  6  feet  long ;  how  wide  is  it  ? 

2.  There  are  two  similar  rectangles  ;  one  is  5  feet  long  and 
2  feet  wide ;  the  other  is  11  feet  long ;  how  wide  is  it  ? 

3.  There  are  two  similar  right-angled  triangles ;  in  the 
largest  the  base  is  9 ;  the  perpendicular  4 ;  in  the  smallest  the 
base  is  8  ;  how  long  is  the  perpendicular  ? 

We  will  now  come  to  the  comparison  of  the  areas  of 
similar  surfaces.     . 

4.  There  are  two  circles ;  one  is  1  foot  in  diameter,  the 
other  2  feet ;  how  much  greater  is  the  area  of  the  larger,  than 
the  area  of  the  smaller  ? 


SIMILAR   FIGURES.  109 

It  is  clearly  more  than  twice  as  large,  for  you  could  lay 
two  of  the  smaller  circles  on  the  larger,  and  still  leave  a 
considerable  space  uncovered.  Before  answering  this  ques- 
tion,, we  will  take  the  simple  case  of  two  squares,  one  of 
which  measures  1  foot  on  a  side,  and  the  other  2  feet.  You 
perceive  the  larger  one  is  4  times  as  great  as  the  smaller. 

Let  one  square  measure  2,  feet  on  a  side ;  the  other,  4  feet ; 
how  much  greater  is  the  larger  than  the  smaller  ?  The 
smaller  contains  4  square  feet ;  the  larger,  16 ;  it  is  therefore 
4  times  as  large. 

5.  If  one  square  measures  3  times  as  much  on  a  side  as 
another,  how  much  greater  is  its  area  than  that  of  the  smaller  ? 

Let  one  square  measure  1  foot  on  a  side ;  the  other,  3  feet ; 
in  what  ratio  are  their  areas  ?  Let  one  measure  2  feet  on  a 
side  ;  the  other,  6  ;  in  what  ratio  are  their  areas  ? 

The  area  of  the  larger  you  find  is  9  times  as  great  as  that 
of  the  smaller.  This  may  serve  to  suggest  the  principle  by 
which  the  areas  of  all  similar  surfaces  may  be  compared. 

The  areas  of  similar  surfaces  are  to  each  other  as  the  squares 
of  their  corresponding  dimensions. 

Let  one  square  measure  1  foot  on  a  side ;  another,  2  feet. 
12;  22::  1:4. 

Let  one  square  measure  2  feet,  and  another,  4  feet  on  a  side. 
22:  42::  4:  16. 

Let  one  square  measure  1  foot  on  a  side  ;  another,  3  feet. 
12:  32::  1:9. 

Let  one  square  measure  2  feet  on  a  side ;  another,  6  feet. 
22  :  62  ::4:36. 

This  principle  applies  to  circles,  triangles,  and  all  s^^ilar 
surfaces  whatever.  You  can  now  recur  to  question  4,  and 
find  the  answer  to  it. 

6.  There  are  two  circles  ;  the  diameter  of  the  greater  is  3 
times  that  of  the  smaller ;  the  area  of  the  smaller  is  1  acre ; 
what  is  tlie  area  of  the  greater  ? 

7.  There  are  two  circles  ;  the  diameter  of  the  smaller  is  2 
thirds  that  of  the  greater  ;  and  the  area  of  the  smaller  is  4 
acres  ;  what  is  the  area  of  the  greater  ? 

8.  There  are  two  similar  triangles ;  «the  corresponding 
dimensions  ar^  as  3  to  4,  and  the  greater  contains  1 2  acres  ; 
what  does  the  smaller  contain  ? 

<  ]0 


no  •  MENTAL   ARITHMETIC. 

9.  A  farmer  fenced  a  triangular  piece  of  ground  for  a  field, 
but  finding  it  not  large  enough,  he  enlarged  it,  making  each 
side  1  third  greater  than  before,  and  it  then  contained  5  acres ; 
how  much  did  it  contain  at  first  ? 

10.  There  is  an  irregular  field  containing  8  acres  ;  one  of 
the  sides  measures  20  rods  ;  if  the  field  be  enlarged,  retain- 
ing the  same  form,  so  that  the  above  named  side  measures  25 
rods,  how  much  land  will  it  contain  ? 

11.  There  are  2  circles  ;  the  smaller  is  3  rods,  the  latter  7 
rods  in  diameter ;  how  much  greater  in  proportion  is  the  area 
of  the  latter  than  that  of  the  former  ? 

12.  There  are  2  circles,  one  with  a  diameter  of  3  feet,  the 
other  of  8  ;  how  much  greater  is  the  area  of  the  larger  than 
that  of  the  smaller  ? 

COMPARISON  OF  SIMILAR  SOLIDS. 

We  now  come  to  the  comparison  of  similar  solids. 

1.  Let  there  be  2  cubes;  one  of  them  measuring  1  inch  on 
a  side,  the  other  2  inches ;  how  much  greater  is  one  than  the 
other? 

You  will  perceive,  by  thinking  of  the  construction  of  the 
cube,  that  the  cube  measuring  2  inches  has  in  it  8  cubic 
inches,  and  is  therefore  8  times  as  great  as  the  one  measuring 
only  1  inch. 

2.  Take  cubes  measuring  1  inch  and  3  inches;  how  much 
greater  is  the  latter  than  the  former  ? 

3.  Let  one  measure  1  inch,  the  other  4  inches ;  how  much 
greater  will  the  larger  cube  be  ? 

These  examples  may  suggest  the  principle  on  which  all 
similar  solids  are  compared. 

/Similar  solids  are  to  each  other  as  the  cubes  of  their  correS' 
ponding  dimensions. 

Take  now  the  first  of  the  above  three  examples ;  the  ratio 
of  the  corresponding  dimensions  is  2  :  1,  the  cubes  of  these 
terms,  or,  2^:13  are  8  :  1,  and  this  is  the  proportion  of  the 
one  solid  to  the  other. 

Li  the  second  example,  the  ratio  of  th^  corresponding  di- 
mensions is  3:1,  the  cubes  of  these  3^  :  1^,  are  27  :  1,  and 
this  is  the  ratio  of  the  two  solids. 


COMPARISON    OF    SIMILAR    SOLIDS.  Ill 

In  the  third  example  the  ratio  of  the  corresponding  dimen- 
sions is  4:1;  the  cubes  of  these  terms,  4^  :  13  are  64  :  1, 
which  is  the  ratio  of  the  two  solids  to  each  other. 

4.  There  are  2  iron  balls ;  the  smaller  is  1  inch,  the  other  5 
inches  in  diameter ;  how  much  does  the  larger  weigh  more 
than  the  smaller? 

5.  There  are  2  iron  balls ;  their  diameters  are  2  inches 
and  3  inches  ;  what  is  the  ratio  of  their  weight  ? 

6.  If  the  diameter  of  2  balls  is  respectively  3  inches  and  4 
inches,  what  is  the  ratio  of  their  weight  ? 

7.  If  a  cubic  inch  of  stone  weigh  1  ounce,  how  many  ounces 
would  a  cubic  stone,  measuring  10  inches,  weigh  ? 

8.  How  many  ounces,  if  the  cube  measured  11  inches? 

9.  How  many  ouncps,  if  the  cube  measured  12  inches? 

10.  If  there  were  a  smaller  pyramid,  of  the  same  material 
and  shape  with  the  great  pyramid  of  Egypt,  and  of  ^  its 
height,  how  many  such  would  it  take  to  equal  in  solid  contents 
the  great  pyramid  ? 

11.  A  common  brick  weighs  4  pounds,  and  is  8  inches  in 
length ;  how  much  will  a  similarly -shaped  brick  weigh,  that 
measures  16  inches  in  length? 

12.  If  an  axe  4  inches  wide  weighs  4J  pounds,  what  will 
be  the  weight  of  a  similar  axe  5  inches  wide  ? 

13.  If  a  blacksmith's  anvil,  1  foot  long,  weighs  200  pounds, 
how  much  will  a  similar  anvil  weigh  that  is  2  feet  long  ? 

14.  A  farmer  sells  2  stacks  of  hay  of  the  same  shape  and 
solidity ;  the  smaller  is  10  feet  high,  and  is  found  to  weigh  3 
tons  ;  the  larger  is  15  feet  high ;  how  can  its  weight  be  deter- 
mined without  weighing  it,  and  what  will  the  weight  be  ? 

15.  There  are  2  similar  cisterns,  the  smaller  is  6  feet  deep 
and  holds  500  gallons  ;  the  lai'ger  is  8  feet  deep  ;  how  many 
gallons  will  it  contain  ? 

These  operations  will  be  rendered  more  easy,  if,  in  every 
case  where  the  ratios  may  be  reduced,  you  reduce  them  to 
their  lowest  terms. 

16.  If  a  coal-pit  8  feet  high  has  required  10  cords  of  wood, 
how  much  wood  would  be  required,  for  a  coal-pit  of  similar 
shape  10  feet  high  ? 

17.  If  there  are  two  trees  shaped  alike,  the  smaller  meas- 
uring 4  feet  in  circumference,  the  larger  5,^et,  how  will  the 
amount  of  wood  in  the  one  compare  with  that  in  the  other  ? 


112  MENTAL   ARITHMETIC. 

The  principle  given  above  applies  to  all  similar  solids, 
whether  bounded  by  plain  surfaces,  or  by  curved  surfaces. 

18.  If  a  dwarf  measures  2  feet  in  height,  and  a  man  of  the 
aame  form  and  solidity,  6  feet  high,  weighs  180  pounds,  how 
many  such  dwarfs  would  equal  the  weight  of  the  man  ?  What 
would  the  dwarf  weigh  ? 

19.  If  a  man  6  feet  2  inches  in  height  weighs  200  pounds, 
what  would  be  the  weight  of  a  giant  of  equal  solidity  and 
similar  form,  9  feet  3  inches  in  height  ? 

20.  If  an  animal  4  feet  high  weighs  600  pounds,  what  will 
an  animal  of  the  same  form  and  equal  solidity  weigh,  whose 
height  is  5  feet  ? 

21.  An  artist  in  Europe  has  made  a  perfect  model  of  St. 
Peter's  Church  at  Rome,  representing  every  part  in  exact 
proportion,  on  a  scale  of  1  foot  to  100  feet ;  if  the  material  of 
the  model  is  of  the  same  solidity  with  that  of  the  church,  how 
many  times  greater  is  the  solid  contents  of  the  church  than 
that  of  the  model  ? 

22.  If  a  granite  obelisk  were  constructed  in  the  precise 
form  of  the  Bunker  Hill  monument,  of  one  tenth  its  height, 
how  many  such  obelisks  would  the  monument  furnish  material 
to  constrtict  ? 

The  comparison  of  similar  surfaces  and  solids  by  proportion 
has  various  interesting  applications  in  determining  the  com- 
parative strength  of  timbers  and  materials  used  in  building, 
and  in  other  arts. 

Case  First.  —  The  strength  of  materials  to  resist  a  strain 
lengthwise. 

1.  If  an  iron  rod  half  an  inch  in  diameter  will  hold  a  cer- 
tain weight  suspended  by  it,  how  much  greater  weight  will  a 
rod  hold  that  is  1  inch  in  diameter  ? 

Here  the  strength  is  in  proportion  to  the  size,  without  re- 
gard to  the  length ;  that  is,  as  the  square  of  the  diameters. 

2.  If  an  iron  rod  half  an  inch  in  diameter  will  suspend  2 
tons,  what  weight  will  a  rod  susp^id  that  is  three  fourths  of 
an  inch  in  diameter  ? 

3.  A  builder  finds  that  an  iron  rod  1  inch  in  diameter  will 
suspend  a  certain  weight ;  he  wishes,  however,  to  add  to  the 
weight  half  as  much  more,  and,  in  order  to  support  it,  substi- 


STRENGTH    OF   MATERIALS.  113 

tiites  for  thfi  inch  rod  another  rod  l^^^  inches  in  diameter  j  will 
it  sustain  the  required  weight  ? 

4.  There  are  two  ropes  of  the  same  material ;  one,  1^  inches 
__     in  diameter ;  the  other,  2  inches  ;  what  is  the  ratio  of  their 
strength  ? 

Case  Second.  —  The  strength  of  beams  to  resist  fracture 
crosswise.  In  beams  of  the  same  material,  length  and  width, 
but  of  diflferent  depth,  the  strength  varies,  as  the  square  of  the 
depth, 

1.  There  are  two  beams  of  equal  length,  but  the  depth  of 
one  is  10  inches  ;  of  the  other,  12  inches;  what  is  the  ratio  of 
their  strength  ?  .         ' 

2.  There  is  a  stick  of  timber  4  inches  thick  and  12  inches 
deep  ;  if  sawed  into  three  4  inch  joists,  what  part  of  the  former 
strength  of  the  whole  stick,  when  placed  edgewise,  will  each 
part  possess,  allowing  nothing  for  waste  in  sawing  ? 

3.  There  is  a  stick  of  timber  10  inches  in  depth ;  if  4  inches 
of  its  depth  be  removed,  what  will  be  its  strength  compared 
to  what  it  was  before  ? 

4.  There  are  two  sticks  of  timber,  equal  in  length  and 
width ;  one,  7  inches  deep ;  the  other,  5  ;  what  is  the  ratio  of 
their  strength  ? 

5.  If  a  stick  of  timber  6  inches  deep  have  2  inches  of  the 
depth  removed,  will  it  be  weakened  more  than  one  half? 

What  is  the  exact  ratio  of  its  present,  compared  with  its 
former  strength  ? 

6.  A  builder  went  to  a  lumber-yard,  wishing  to  obtain  an 
oak  beam  5  inches  wide  and  10  inches  deep  ;  the  lumber- 
merchant  said,  "  I  have  not  such  a  stick ;  but  I  have  two  oak 
sticks  of  the  right  length  and  width,  and  7  inches  deep ;  they 
will  both,  placed  side  by  side,  be  stronger  than  one  beam  10 
inches  deep."     "  Not  so  strong,"  said  the  builder. 

Which  was  right  ?  and  what  is  the  ratio  of  strength  in  the 
two  cases  ? 

10* 


NOTES    TO    PART    FIRST. 


Note  1.  —  Page  15. 


This  exercise  should  be  often  reviewed  till  the  pupils  can  gi) 
through  it  with  ease,  and  without  mistake.  No  exercise  can  be 
devised  that  will  more  rapidly  increase  the  learner's  powers  in  Ad- 
dition. 

Note  2.  —  Page  16.     To  the  Instructor. 

The  word  complement  means,  something  to  fill  up.  In  arithme- 
tic, the  complement  of  a  number,  strictly  speaking,  is  that  number 
which  must  be  added  to  it,  to  make  it  up  to  the  next  higher  order. 
The  complement  of  a  number  consisting  of  units  only,  as  3,  7,  9, 
is  the  number  that  must  be  added  to  make  it  up  to  10,  and  con- 
sists of  units  only.  If  the  number  consist  of  tens,  as  20,  50,  its 
complement  is  the  number  that  must  be  added  to  make  a  hundred, 
and  consist  of  tens.  If  the  number  is  hundreds,  its  complement  is 
so  many  hundreds  as  will  make  up  a  thousand. 

If  the  number  consist  of  several  orders,  its  full  complement 
will  consist  of  the  same  orders,  of  such  an  amount  as  to  raise  the 
-gum  to  the  next  order  above  the  highest  named  in  it.  The  com- 
plement of  745  is  255,  for  745-[-255— 1000,  which  is  the  order 
next  above  the  highest  named  In  the  given  sum. 

The  more  restricted  use  of  the  word,  as  employed  in  the  text,  is 
sufficient  for  the  purposes  here  had  in  view. 

A  few  suggestions  will  here  be  made  in  reference  to  the  best 
mode  of  conducting  the  accompanying  recitation.  The  object  of 
the  lesson  is  to  cultivate  the  po^ver  of  instantly  associating  a  num- 
ber and  its  complement  together.  In  conducting  tha  recitation,  the 
answer  to  each  question  as  it  is  given  out,  should  be  required  sim- 
ultaneously by  the  whole  class.  The  teacher  should  stand  before 
them,  and  require  that  every  eye  be  fixed  on  him.  The  questions 
should  not  be  hurried,  but  the  class  should  be  encouraged  to  an- 
swer instantly  on  hearing  the  question.  This  will  be  easy  in  the 
first  class  of  numbers  given,  which  are  even  tens.     In  regard  to 


NOTES    TO    PART    FIRST.  115 

the  remaining  numbers,  however,  which  are  not  even  ti?ns,  some- 
thing more  will  be  necessary.  Suppose  the  question  is,  what  is  the 
complement  of  37  ?  it  may  be  conducted  as  follows : 

Teacher.     What  is  the  complement  of —  30  ? 

Class^     70.  ^ 

Teacher.  Now  listen  to  me  without  speaking ;  what  is  the  com- 
plement of  30 ?  you  observe,  I  am  going  to  say  something 

more ;  what  will  it  be  ? 

Class.     Something  between  30  and  40. 

Teacher.  "Well  then,  whereabouts  will  the  complement  be 
found  ? 

Class.    Between  60  and  70. 

Teacher.  Very  good  !  Now  when  I  say  30,  and  keep  my  voice 
suspended,  showing  that  that  is  not  all,  what  number  can  you  think 
of,  that  you  know  will  be  a  part  of  the  complement  ? 

Class.     60. 

Teacher,  Very  well.  Now  listen ;  what  is  the  complement  of 
80 ?  what  have  you  now  in  your  mind  ? 

Class.     60. 

Teacher.  Well,  now  once  more  listen,  and  all  answer  as  soom 
as  you  hear  the  question ;  what  is  the  complement  of  3  7  ? 

Class.     63. 

In  the  following  questions,  let  the  teacher  always  make  a  short 
pause  between  pronouncing  the  tens,  and  the  units ;  and  if  the 
class  hesitate  or  disagree  in  their  answer,  let  the  question  be  re- 
solved into  its  elements,  and  each  one  presented  separately.  Thus, 
if  64  is  the  number,  and  the  class  have  not  answered  promptly 
and  alike,  say  thus,  —  what  is  the  complement  of  60  ? 

Class.     40. 

Teacher.    What  is  the  complement  of  60 ■  ?  what  do  you 

think  of? 

Class.     30. 

Teacher.  Now  answer  all  together ;  what  is  the  complement  of 
64  ? 

Class.     36. 

In  the  examples  of  addition  that  follow,  the  teacher  should  make 
%  pause  between  the  two  numbers,  and  see  that  every  member  of 
the  class  is  intent  and  eager  to  catch  the  second  number,  and  an- 
swer instantly.  A  few  questions  answered  by  the  whole  class  in 
this  way,  will  benefit  them  more  than  whole  pages  recited  in  an 
indolent  and  listless  manner. 


Note  3.  —  Page  17. 

In  these  and  all  other  examples,  the  large  numbers  should  be 
taken  first.  If  the  pupil  begins  with  the  units,  as  in  written 
arithmetic,  he  should  be  checked  at  once.     Such  a  method  would 


116  NOTKS     TO     PART    FIRST. 

only  lead  to  a  laborious  imitation  of  the  j)rocess  of  written  arith- 
metic, which  is  not  the  natural  one,  and  could  give  no  new  power 
to  the  pupil,  nor  awaken  any  new  interest  in  the  study.  Only  a 
small  portion  of  these  questions  should  be  recited  at  one  lesson. 


Note  4.  — Page  23. 

Care  must  be  taken  here  that  the  pupil  does  not  imitate  the 
process  of  written  arithmetic,  but  be  required  to  regard  every 
number  in  its  true  value.  Thus  in  the  question,  what  is  one  fifth 
of  250  1  he  must  not  say  5  In  25  is  contained  5  times:  and  5  in  0,  no 
times ;  but  one  fifth  of  25  is  5  ;  therefore,  one  fifth  of  250  is  50. 


Note  5. — Page  25. 

In  the  higher  as  well  as  the  lower  numbers,  let  the  pupil  grapple 
at  once  with  the  number  as  It  stands.  In  this  way  his  interest  will 
be  very  much  increased.  He  will  see,  throughout,  the  progress  he 
is  making;  whereas,  in  written  arithmetic  as  usually  studied,  the - 
pupil  has  no  sooner  begun  an  operation  than  he  loses  sight  of  the 
process,  and  goes  on  in  blind  bondage  to  his  rule,  till  he  comes  out 
at  the  end,  and  then  looks  to  the  book,  as  to  an  oracle,  for  the 
answer. 

Let  the  oldest  class  in  arithmetic  In  a  school  be  called  up,  and 
one  of  them  be  required  to  perform  on  the  board  the  question, 
*'  what  is  one  sixth  of  43,248  ?  "  and  when  he  has  obtained  the  first 
quotient  figure,  stop  him,  and  ask  him,  what  he  has  now  done ;  he 
will  most  likely  be  unable  to  tell.  The  answer  he  will  give- will 
probably  be,  that  he  has  divided  43  by  G  ;  and  no  one  of  his  class 
will  probably  have  a  better  answer  to  offer.  If  he  -  says  he  has 
divided  43  thousand,  he  is  still  wrong ;  for  he  has  divided  only  42 
thousand,  leaving  one  thousand  undivided. 

In  some  of  the  examples  given  in  this  section,  the  large  num- 
bers may  be  separated  in  diiferent  ways  preparatory  to  division. 
Thus,  in  the  last  example,  92,648  may  be  divided  80,000,  12,000, 
600,  48 ;   or  88,000,  4000,  640,  8,  and  in  still  other  ways. 

Pupils  should  be  encouraged  to  exhibit  more  methods  than  one 
for  obtaining  the  answer.  If  a  scholar  has  two  methods  he  should 
be  allowed  to  give  them  both,  and  if  another  has  a  different  one 
still,  it  should  be  brought  forward,  and  the  most  lucid  and  easy  one 
should  receive  the  commendation  of  the  teacher. 

Note  6.  —  Page  81.  Strictly  speaking,  there  is  no  relation  in 
quantity  between  a  line  and  a  surface,  but  only  between  a  line  and 
the  dimensions  of  a  surface.  By  the  square  of  a  line  is  meant  a 
square  surface,  each  of  whose  sides  has  th^  same  length  as  the 
given  line. 


PART     SECONDi 


CONTAINING 


RULES   AND   EXAMPLES    FOR   PRACTICE      , 

IN 

WRITTEN    ARITHMETIC. 


NUMERATION  OF  WHOLE  NUMBERS. 

In  common  Arithmetic  there  are  9  figures  used  for  the  ex- 
pression of  numbers.  1,  one;  2,  two;  3,  three ;  4,  four;  5, 
five ;  6,  six  ;  7,  seven ;  8,  eight ;  9,  nine.  jpSVhen  one  of  these 
figures  stands  alone,  it  signifies  so  many  units,  or  ones  ;  when 
two  figures  stand  side  by  side,  the  left  hand  figure  signifies  so 
many  tens ;  when  three  stand  side  by  side,  the  left  hand  figure 
signifies  so  many  hundreds ;  and  universally,  as  you  advance 
to  the  left,  the  figures  increase  in  value  tenfold  at  each  step, 
as  will  be  seen  in  the  table  on  the  next  page. 

The  right  hand  place  is  always  that  of  units.  When  there 
are  tens,  and  no  units,  a  cipher,  0,  must  stand  in  the  unit's 
place,  thus,  20 ;  this  merely  serves  to  occupy  the  unit's  place, 
and  shows  that  the  figure,  2,  is  in  the  place  of  tens.  When  there 
are  hundreds,  and  no  tens  nor  units,  two  ciphers  are  wanted ; 
one  in  the  unit's  place,  and  one  in  the  place  of  tens ;  as,  200  ; 
and  so  of  all  higher  numbers. 

To  annex  a  cipher  to  a  figure,  therefore,  is  the  same  as  to 
multiply  the  number  by  ten,  for  it  removes  the  figure  from 
the  unit's  place  to  the  place  of  tens.  To  annex  two  ciphers 
is  the  same  as  to  multiply  the  number  by  a  hundred,  for  it 
removes  the  figure  from  the  unit's  place  to  that  of  hundreds. 


118 


TABLE    OF   NUMERATION. 


two. 

two  tens,  that  is,  twenty. 

two  hundred. 

enumerate.    • 

enumerate. 

f  two  tens  of  thousands, 
( that  is,  twenty  thousand. 

two  hundred  thousand. 

enumerate. 

twenty-two  million. 

enumerate. 

enumerate. 


In  writing  numbers,  every  place  not  occupied  by  a  figure 
must  be  occupied  by  a  cipher ;  otherwise  the  true  value  of  the 


NUMERATION. 


119 


figures  at  the  left  hand  of  that  place  woqld  not  be  preserved. 
Thus,  if  you  wish  to  write  in  figures  the  number,  three  hun- 
dred and  four,  as  there  are  no  tens,  a  cipher  must  stand  in  the 
place  of  tens,  304.  Should  you  omit  the  cipher,  and  write  34, 
the  3  would  have  slid  into  the  ten's  place,  and  it  would  not 
express  three  hundred  and  four. 

As  in  advancing  to  the  left,  figures  increase  their  value 
tenfold  at  each  step,  so  if  you  begin  at  any  place  in  a  line  of 
figures,  and  move  towards  the  right,  the  figures  will  diminish 
in  value  tenfold  at  each  step.  That  is,  each  figure  will  sig- 
nify but  a  tenth  part  of  what  it  would,  if  it  stood  in  the  next 
left  hand  place.     This  will  prepare  you  to  look  at  the 


Whole  Numbers. 


NUMERATIOlil   OF  DECIMALS. 
Decimals. 


^S 


I 
3 


two,  and  two  tenths. 
I  four,  and  two  tenths,  and  fivo 
I    hundredths,  or  25  hundredths. 

( twenty-two    and    twenty- 
(      two  hundredths. 

two  thousandths, 
enumerate. 

{four  hundred  and  seventeen  hun- 
dred thousandths.  . 

enumerate. 


120 


ADDITION. 


SECTION    I. 


ADDITION. 

Addition  is  the  uniting  of  several  sums  into  one,  to  show 
their  amount. 

Rule.  Set  down  the  numbers,  units  under  units,  tens  under 
tens,  and  so  on.  Add  the  column  of  units,  set  down  the  units 
of  the  amount,  and  carry  the  tens,  if  there  are  any,  to  the 
column  of  tens ;  add  the  column  of  tens,  and  set  down  the  unit 
figure  of  the  amount,  carrying  the  figure  of  tens  to  the  next 
column  ;  and  so  on.  In  adding  the  last  column  set  down  the 
whole  amount. 

To  prove  the  work,  repeat  the  operation,  beginning  at  the 
top  and  adding  downwards. 


1.  472+842. 

2.  376+421-}-645. 

3.  431-}-843+794. 

4.  821+954+359. 

5.  267+549+121. 

6.  834+682+762. 

7.  468+912+683. 

8.  871+934+340. 

9.  516+617+713. 

10.  685+937+742. 

11.  840+931+672. 

12.  963+847+784. 

13.  421+317+844. 


Examples. 

14.  6342+1896+4741+8962. 

15.  3249+856+8007+4990. 

16.  3819+42+906+1728. 

17.  1645+2718+92+1807. 

18.  1543+1899+3054+26. 

19.  1854+1962+2168+666. 

20.  1062+6300+9071+7001. 

21.  2593+1801+9201+2113. 

22.  9064+2118+1802+3076. 

23.  1001+9016+7990+26. 

24.  106+2307+9436+108. 

25.  1214+6403+7113+4009. 


26.  In  1840,  the  population  of  the  New  England  States 
was  as  follows :  Maine,  501,793  ;  New  Hampshire,  284,574 ; 
Vermont,  291,948 ;  Massachusetts,  737,699 ;  Connecticut, 
309,978  ;  Rhode  Island,  108,850.  What  was  the  population 
of  all  the  New  England  States  ? 

27.  The  population  of  the  Middle  States,  in  1840,  was  as 
follows :  New  York,  2,428,921 ;  New  Jersey,  373,306  ;  Penn- 
sylvania, 1,724,033  ;  Delaware,  78,085  ;  Maryland,  469,232  ; 
Virginia,  1,239,797.  What  w^as  the  total  population  of  the 
Middle  States  ? 


ADDITION.  121 

28.  The  population  of  the  Southern  States,  in  1840,  was : 
North  Carolina,  753,419  ;  South  Carolina,  594,398  ;  Georgia, 
691,392 ;  Alabama,  ,590,756 ;  Tennessee,  829,210 ;  Mississippi, 
375,651 ;  Ai'kansas,  97,574 ;  Louisiana,  352,411.  What  was 
the  total  population  of  these  States  ? 

29.  In  1840,  the  population  of  the  Western  States  was  as 
follows  :  Ohio,  1,519,467  ;  Indiana,  685,866 ;  Illinois,  476,183  ; 
Michigan,  212,267  ;  Kentucky,  777,828  ;  Missouri,  383,702. 
What  was  the  total  population  of  these  States  ? 

30.  What  is  the  total  population  of  all  the  United  States, 
as  set  down  in  the  four  preceding  examples  ? 

Whenever,  in  adding  a  column,  two  figures  occur  together, 
which  amount  to  10,  as  8  and  2,  7  and  3,  take  them  both  to- 
gether and  cal?  them  10.  This  will  make  the  addition  more 
rapid  and  easy. 

When  you  have  become  familiar  with  the  operations  in  ad- 
dition, you  may  occasionally  vary  your  method,  by  taking  two 
columns  of  figures  at  a  time.  If  you  have  been  thorough  in 
the  mental  part  of  this  wock,  you  will  be  able  to  do  this.  It 
will  furnish  an  agreeable  variation  in  your  method  of  work, 
and  greatly  increase  your  power  of  rapid  calculation. 

31.  This  method  is  seen  in  the  following  example  : 


3124 
7681 
4942 

L  15747 


42  and  81  are  123,  and  24  are  147  ;  set  down 
the  47,  and  carry  the  1  hundred  to  the  column 
of  hundreds ;  50  and  76  are  126,  and  31  are 
157. 

It  will  be  well  often  to  adopt  this  as  your  method  of  proof. 
After  performing  the  work  by  taking  one  column  at  a  time, 
prove  it  by  taking  two  columns ;  or  perform  it  first  in  the  latter 
way,  and  prove  it  in  the  other. 

32.  1467+894+1721+8396. 

33.  9461+8134+2016+4317. 

34.  84161+9632+78167+43180. 

35.  109761+20671+437674+963. 

36.  26431+184097+467124+84321. 

37.  43126+91434+237210+127. 

38.  1235467+1096+34271+4081. 

39.  10467+31762+10921+9634. 

40.  37193+10634+206721+104367. 

11      - 


122 


SUBTRACTION. 


SECTION    II. 


SUBTRACTION. 

Subtraction  is  the  taking  of  a  smaller  number  from  a 
larger,  to  show  the  difference.  The  larger  number  is  called 
the  minuend ;  the  smaller,  the  subtrahend ;  the  diflference  is 
called  the  remainder. 

Rude.  Set  down  the  numbers,  the  larger  number  upper- 
most, units  under  units,  tens  under  tens.  Subtract  the  units 
of  the  lower  number  from  the'  unit  figure  above,  and  set  down 
the  difference.  Proceed  in  the '  same  way,  with  the  tens  and 
higher  orders,  to  the  close.  If,  in  any  case,  the  figure  of  the 
minuend  is  less  than  the  figure  below  it,  increase  it  by  ten,  by 
borrowing  one  from  the  next  higher  figure  of  the  minuend, 
remembering  at  the  next  step,  that  the  figure  in  the  minuend 
has  already  been  diminished  by  1. 

To  prove  the  work,  add  the  remainder  and  the  subtrahend 
together,  and,  if  the  work  is  correct,  the  sum  will  agree  with 
the  minuend. 

Examples. 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 


748—365. 

674—582. 

849—634. 

347—267. 

431—249. 

867—312. 

419—224. 

519—499. 

318—201. 

856—106. 

3416—2999. 

4162—4091. 

7089—3007. 


14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 


8990—7096. 

8243—6492. 

784—96. 

210—100. 

681—504. 

901—75. 

16432—14968. 

195864—137461. 

228476—13962. 

740016—116799. 

86400—199. 

10006—4364. 


26.  America  was  discovered  in  1492 ;  Plymouth  was  settled 
in  1620  ;  how  long  was  that  after  the  discovery  of  America? 

27.  The  Independence  of  the  United  States  was  declared 
in  1776 ;  how  long  was  that  after  the  settlement  of  Plymouth  ? 

28.' George  Washington  was  born  in  1732;  betook  com- 
mand of  the  American  armies  in  1776 ;  how  old  was  he  then  ? 

29.  Gen.  Washington  became  President  of  the  United 
States  in  1789  ;  how  old  was  he  then  ? 


SUBTRACTION.  -  123 

30.  In  1820,  the  population  of  Maine  was  298,335  ;  in  1830 
it  was  399,955  ;  what  was  the  increase  in  10  years  ? 

dl.  The  population  of  Maine  in  1840,  was  501,973  ;  what 
was  the  increase  fi'om  1830  to  1840  ? 

32.  The  population  of  Massachusetts  in  1810,  was  472,040, 
in  1820,  523,487  ;  how  much  had  it  increased  from  1810  to 
1820  ? 

33.  The  population  of  Massachusetts  in  1830,  was  610,408 ; 
how  much  had  it  increased  from  1820  to  1830  ? 

34.  In  1840,  the  "population  of  Massachusetts  was  737,699  ; 
how  much  had  it  increased  from  1830  to  1840  ? 

35.  The  population  of  the  §tate  of  New  York  in  1810,  was 
959,949  ;  in  1820,  it  was  1,372,812  ;  what  was  the  gain  ? 

36.  The  population  of  New  York  in  1830,  was  1,918,608 ; 
what  was  the  gain  from  1820  to  1830  ? 

37.  In  1840,  it  was  2,428,921 ;  what  was  the  gain  from 
1830  to  1840  ? 

38.  The  population  of  Ohio  in  1810,  was  230,760  ;  in  1820 
it  was  581,434  ;  what  was  the  gain  from  1810  to  18*20  ? 

39.  In  1830,  the  population  of  Ohio  was  937,903  ;  what  was 
the  increase  from  1820  to  1830  ? 

40.  In  1840,  the  population  of  Ohio  was  1,519,467 ;  what 
was  the  increase  from  1830  to  1840  ? 

Another  method  of  performing  subtraction,  often  more  con- 
venient than  the  former,  is  the  following : 

Regard  the  subtrahend  as  a  round  number,  one  greater  than 
the  figure  of  its  highest  order ;  that  is,  if  the  subtrahend  is  43, 
call  it  50  ;  if  251,  call  it  300  ;  subtract  this  round  number  from 
the  minuend,  and  then  to  the  remainder  add  the  complement 
required  to  make  up  the  subtrahend  to  the  round  number ;  as 
follows : 

41.  674—381 ;  400  from  674  leaves  274 ;  add  19,  the  com- 
plement of  381  ;  274+19=293,  ans. 

Apply  this  method  to  example  11  above. 

42.  3416 — 2999  ;  the  first  remainder  you  see  is  416,  and 
the  complement  is  1.  Ans.  417. 

This  example  shows  how  much  shorter  the  work  often  be- 
comes by  adopting  this  method. 

One  of  the  above  methods  may  be  used  as  a  proof  of  the  other. 

43.  384—219.  I      45.  1679—291. 

44.  1260—984.  46.  2496—954. 


124 


MULTIPLICATION. 


SECTION    III 


MULTIPLICATION. 

In  multiplication  a  number  is  repeated  a  certain  number  of 
times,  and  the  result  thus  obtained  is  called  the  Product. 

Rule. 

Set  down  the  smaller  factor  under  the  larger,  units  under 
units,  tens  under  tens.  Begin  with  the  unit  figure  of  the 
multiplier ;  multiply  by  it,  first,  the  units  of  the  multiplicand, 
setting  down  the  units  of  the  product,  and  reserving  the  tens 
to  be  added  to  the  next  product.  Proceed  thus  through  all 
the  figures  of  the  multiplicand.  If  there  are  more  figures 
than  one  in  the  multiplier,  take,  next,  the  tens,  and  multiply 
the  figures  of  the  multiplicand  as  before,  setting  the  figures  of 
the  product  one  degree  farther  to  the  left  than  before. 

Add  the  several  partial  products,  and  the  amount  Avill  be 
the  whole  product. 

Examples. 

342]  Proof, 

64 

The  best  practical  meth- 
od of  proof  is  carefully 
to  repeat  the  opera- 
tion. 


1.  342X64.    Thus, 


1368 
.2052 


21888     Ans. 


2.  346X34. 

3.  579X82. 

4.  976X38. 

5.  826X91. 

6.  376X121. 

7.  345X243. 

8.  798X114. 

9.  6181X35. 

10.  6821X82. 

11.  7413X96. 

12.  7921X22. 

13.  8964X85. 

14.  9056X43. 

15.  8007X41. 

16.  4559X741. 


17.  9642X864. 

18.  8721X317. 

19.  1841X134. 

20.  13763X26. 

21.  97623X318. 

22.  1172671X216. 

23.  1874215X341. 

24.  742634X912. 

25.  189423X62. 

26.  14376281X194. 

27.  17284265X36. 

28.  671234X427. 

29.  1895453X28. 

30.  3469528X672. 

31.  906421384X923. 


MtJLTlt^LlCATlON.  125 


35.  443754262X916. 

36.  1123496113X413. 


32.  713489605X84. 

33.  843469537X906. 

34.  236749024X516. 

37.  The  average  length  of  the  State  of  Massachusetts  ia 
150  miles  its  breadth,  50  miles ;  how  many  square  miles  does 
it  contain  ? 

38.  The  average  length  of  Pennsylvania  is  275  miles ;  its 
breadth,  1 65  miles ;  how  many  square  miles  does  it  contain  ? 

39.  The  State  of  Ohio  averages  223  miles  in  length,  180  in 
breadth  ;  how  many  square  miles  does  it  contain  ? 

40.  The  StaM  of  Illinois  averages  245  miles  in  length,  147 
in  breadth ;  how  many  square  miles  does  it  contain  ? 

41.  If  there  are  365  days  in  one  year,  how  many  days  are 
there  in  25  years  ? 

42.  If  the  wages  of  a  soldier  is  8  dollars  a  month,  what  will 
be  the  wages  of  7867  soldiers  for  12  months  ? 

43.  There  are  320  rods  in  1  mile ;  how  many  rods  are  there 
in  278  miles  ? 

44.  741X84.  47.  946734X496. 

45.  19643X892.  48.  1623X198. 

46.  246731X9210.  49.  9336X1998. 

When  the  multiplier  is  a  composite  number,  you  may  mul- 
tiply first  by  one  of  its  factors,  and  the  product  thus  obtained 
by  the  other  factor,  or  by  the  others  in  succession  if  there  are 
more  than  two. 

Apply  this  method  to  the  following  examples :  — 

50.  8476X45.         51.  1371X125.         52.  7465X108. 

If  a  figure  in  the  multiplier  is  a  factor  of  the  figure  in  the 
next  higher  place,  you  may  shorten  the  operation  by  multi- 
plying the  partial  product  of  the  lower  figure  by  the  other 
factor  of  the  higher:  thus,  in  Ex.  44,  above,  having  found 
4  times  741,  you  know  that  8  times  the  same  is  twice  as  many, 
and  80  times  is  20  times  as  many ;  you  need,  therefore,  only 
double  the  line  of  the  first  partial  product,  setting  it  one  de- 
gree farther  to  the  left,  to  express  the  tenfold  higher  value. 
The  same  may  be  done  if  the  right  hand  figure  is  a  factor  of 
the  number  expressed  by  the  next  two  higher  figures. 

Apply  the  process  to  the  following  examples :  — 

53.  947X639.  56.  27934x369. 

•    54.  13674X4812.  57.  67514x64164. 

55.  19742X568.  58.  259385X13212. 

11* 


126  DIVISION. 

If  the  multiplier  is  10,  or  any  power  of  10,  annex  to  the 
multiplicand,  for  the  answer,  as  many  ciphers  as  there  are  in 
the  multiplier. 

If  the  multiplier  consists  of  9s,  add  as  many  ciphers  to  the 
multiplicand  as  there  are  9s  in  the  multiplier,  and  from  the 
product  subtract  the  multiplicand.  The  remainder  will  be 
the  product  sought ;  for,  by  adding  the  ciphers,  you  multiply 
by  a  number  greater  by  one  than  the  multiplier.  The  multi- 
plicand, therefore,  will  be  found  in  the  product  once  too  many 
times.  So  if  the  multij^lier  is  2  or  3  less  than  some  power 
pf  10,  you  may  do  the  same,  remembering  to  take  the  multi- 
plicand out  as  many  times  as  the  multiplier  is  units  less  than 
a  power  of  10. 

In  this  way  perform  the  following  examples :  — 

59.  3847X99.  61.  54327X98. 

60.  4572X999.  62.  45314X997. 


SECTION    IV. 

DIVISION. 

In  Division,  two  numbers  are  given,  in  order  to  find  how 
many  times  one  contains  the  other ;  or,  in  order  to  separate  one 
number  into  as  many  equal  parts  as  there  are  units  in  the  other. 

The  number  to  be  divided  is  the  dividend ;  the  number  it 
is  divided  by  is  the  divisor  ;  the  answer  is  the  quotient. 

To  perform  the  operation,  set  down  the  divisor  •  at  the  left 
of  the  dividend.  Take  as  many  figures  on  the  left  of  the 
dividend  as  will  contain  the  divisor  one  or  more  times.  See 
how  many  times  the  divisor  is  contained  in  these  figures,  and 
set  down  the  number  as  the  first  figure  of  the  quotient. 
Multiply  the  divisor  by  the  quotient  figure,  and  subtract  the 
product  from  the  number  taken.  To  the  remainder  bring 
down  another  figure  of  the  dividend,  and  proceed  as  before. 

1.  13276-r-122 ;  thus,         122)13276(108  quotient. 

122 

1076 
976 

100    remainder. 


DIVISION. 


127 


To  prove  the  work,  multiply  the  divisor  and  the  quotient 
together,  and  add  the  remainder,  if  there  be  any,  and  the 
amount,  if  the  worjc  be  right,  will  be  equal  to  the  dividend. 

Thus  in  the  above  example,         122 

108 


976 

122 

100 

13276 

2. 

11764-^-34. 

11.  3059-r-214. 

3. 

47478-^-82. 

12.  700601-r-34. 

4. 

37088-^38. 

13.  643817-r-150. 

5. 

75116-^-91. 

14.  300796-^145. 

6. 

45496^121. 

15.  3264291-i-27. 

7. 

83835-^-243. 

16.  18947633  :  181 

8. 

90972H-114. 

17.  384628910-^-26. 

9. 

89743-^-17. 

18.  137900-^-62. 

10. 

7426831H-141. 

19.  3946908-^172. 

20.  A  man  divided  35,785  dollars  equally  among  five  chil- 
dren ;  how  much  did  each  receive  ? 

21.  In  one  barrel  of  flour  there  are  196  lbs. ;  how  many 
barrels  of  flour  are  there  in  13,916  lbs.  ? 

22.  In  1840  the  population  of  Maine  was  501,793  ;  the 
State  contained  then  30,000  sq.  miles ;  how  many  inhabitants 
were  there  on  an  average  to  a  sq.  mile  ? 

23.  The  State  of  Massachusetts  contained,  in  1840,  737,699 
inhabitants  ;  its  territory  is  7500  square  miles ;  how  many 
inhabitants  are  there  to  a  square  mile  ? 

24.  The  population  of  Ohio  in  1840  was  1,519,467;  its 
territory  is  40,000  square  miles ;  how  many  inhabitants  to 
a  sq.  mile  ? 

If  the  divisor  is  less  than  12  the  multiplication  and  subtrac- 
tion may  be  carried  on  in  the  mind,  and  only  the  quotient  set 
down.  This  may  most  conveniently  be  written  directly  under 
the  dividend. 

25.  7846-r-3.     Operation,     3)7846 


26.  964385-f-5. 


2615-J-l     Rem. 


128  DIVISION. 


27.  346218-7-7. 

28.  214681-^9. 

29.  684219-i-8. 

30.  9640279-^-4. 


31.  146710063-^-6. 

32.  1143762-i-ll. 

33.  1964217-i-12. 

34.  4691382-^4. 


It  is  welt  to  adopt  the  method  of  short  division  sometimes, 
when  the  divisor  is  larger  than  12. 


35.  33467-H15. 

36.  46943-f-15. 

37.  81743-T-16. 


38.  91674-r-21. 

39.  673845-T-22. 


Miscellaneous  Examples  on  the  foregoing  Rules. 

1.  A  merchant  began  to  trade  with  4325  dollars;  he  gained 
in  one  year  784  dollars  ;  what  was  he  then  worth.  ? 

2.  A  man's  income  is  948  dollars  a  year ;  his  expenses  are 
762  dollars ;  how  much  does  he  save  of  his  income  in  one 
year? 

3.  How  much  will  he  save  in  9  years  ? 

4.  A  man  bequeathed  his  property,  3882  dollars,  one  third 
to  his  wife,  and  the  remainder  in  equal  shares  to  his  four 
children ;  what  was  each  child's  share. 

5.  A  merchant  buys  643  barrels  of  flour,  at  5  dollars  a 
barrel;  he  pays  in  addition — for  freight,  65  dollars,  for  insur- 
ance, 17  dollars  ;  what  does  the  whole  cost  him  then  ?  What 
does  each  barrel  cost  him  ? 

6.  A  drover  bought  7  oxen  for  46  dollars  a  head,  12  cows 
for  32  dollars  a  head,  96  sheep  for  3  dollars  a  head ;  how 
much  did  they  all  come  to  ? 

7.  A  drover  buys  48  head  of  cattle  at  32  dollars  a  head ; 
the  whole  expense  of  driving  them  to  market  and  selling  them 
is  72  dollars  ;  he  sells  them  for  38  dollars  a  head ;  what  does 
he  gain  ? 

8.  A  laborer  receives  16  dollars  for  every  four  weeks' 
labor  ;  he  works  48  weeks ;  what  will  his  earnings  amount  to  ? 

9.  A  man  buys  7  tons  of  hay  in  the  field  for  13  dollars  a 
ton ;  the  cost  of  carrying  it  all  to  market  is  48  dollars  ;  he 
sells  it  for  15  dollars  a  ton ;  does  he  gain  or  lose,  and  how 
much?  "^ 

10.  A  man  receives  a  salary  of  950  dollars  ;  he  spends  for 
groceries  154  dollars,  for  milk  21  dollars,  for  meat  75  dollars  ; 
for  wood  67  dollars,  for  clothing  184  dollars,  for  horse  hire 
38  dollars,  for  journeying  93  dollars,  for  repairs  19  dollars, 


REDUCTION.  1^ 

for  hired  help  132  dollars,  for  attendance  of  the  physician 
26  dollars,  for  furniture  51  dollars,  for  house-rent  184  dollars, 
and  86  dollars  in  churity  and  other  incidental  expenses ;  has 
he  spent  more  than  his  salary,  or  less,  and  how  much  ? 


SECTION    V. 

REDUCTION. 

The  object  in  Reduction  is  to  change  a  quantity,  in  one 
denomination,  to  another,  which  shall  have  the  same  value. 
See  Sec.  VI.,  Part  I.  Higher  denominations  are  reduced  to 
lower  by  multiplication. 

Examples. 

1.  Reduce  3  yds.  to  feet. 

2.  Reduce  42  yds.  to  feet. 

3.  Reduce  4  feet  to  inches.  .     ^^ 

4.  Reduce  17  feet  to  inches.  r  - 

5.  Reduce  132  feet  to  inches.  ^  . 

6.  Reduce  16  yds.  to  inches. 

7.  Reduce  21  yds.  to  inches. 

8.  In  112  feet  7  inches,  how  many  inches? 

9.  In  165  feet  4  inches,  how  many  inches'? 

10.  In  5  yds.,  2  feet,  9  inches,  how  many  inches  ? 

11.  In  24  rods,  how  many  feet  ? 

12.  In  87  rods,  how  many  feet  ? 

13.  In  567  rods,  how  many  inches  ? 

14.  In  7  rods,  4  feet,  how  many  feet? 

15.  In  31  rods,  2  feet,  6  inches,  how  many  inches  ? 

16.  In  131£  how  many  shillings? 

17.  Reduce  781£  to  shillings. 
18.-Reduce  758£  to  shillings. 

19.  Reduce  19  shillings  to  pence. 

20.  Reduce  7£  11  shillings  to  pence. 

21.  Reduce  141£  16  shillings,  4  pence,  to  pence. 

22.  Reduce  4£  7  shillings,  3  pence,  to  farthings. 

23.  Reduce  14  lbs.  8  oz.  Av.  to  oz. 

24.  Reduce  3  qrs.  9  lbs.  13  oz.  to  oz.  -  "* 


130  REDUCTION. 

25.  Reduce  44  cwt.  3  qrs.  19  lbs.  to  lbs. 

26.  Reduce  13  T.  12  cwt.  2  qrs.  to  lbs. 

27.  Reduce  3  lbs.  6  oz.  17  dwt.  Troy,  to  dwt. 

28.  Reduce  13  lbs.  2  oz.  14  dwt.  to  dwt. 

29.  Reduce  4  oz.  16  dwt.  to  grs. 

30.  Reduce  6  lb.  7  oz.  9  dwt.  4  grs.  to  grs. 

31.  Reduce  27  gallons,  wine  measure,  to  pints. 

32.  Reduce  7  hhd.  13  galls.  2  qts.  to  qts. 

33.  Reduce  1  hhd.  to  gills. 

34.  Reduce  174  bushels  to  qts. 

35.  Reduce  73  bushels  to  pints. 

36.  Reduce  231  bushels  to  qts. 

37.  In  13  sq.  feet,  how  manj  sq.  inches? 

38.  In  84  sq.  rods,  how  many  sq.  feet? 

39.  Reduce  13  sq.  rods  to  inches. 

40.  Reduce  3  R.  17  rods,  to  feet. 

41.  Reduce  5  A.  2  R.  14  rods,  to  feet. 

42.  Reduce  1 7  solid  feet  to  inches. 

43.  Reduce  19   s.  yds.  14  feet,  to  inches. 

44.  Reduce  24  s.  yds.  8  feet,  504  inches,  to  inches. 

45.  Reduce  6  cords,  13   s.  feet,  to  feet. 

46.  Reduce  27  cords,  28  s.  feet,  to  feet. 

47.  Reduce  45  cords,  13  s.  feet,  to  feet. 

48.  In  75  E.  E.  of  cloth,  how  many  qrs.  ? 

49.  Reduce  78  yds.  3  qrs.  to  qrs. 

50.  Reduce  194  yds.  1  qr.  to  nails. 

51.  Reduce  11  yds.  3  qrs.  2  nails,  to  inches. 

52.  Reduce  174  E.  Fr.  to  nails. 

53.  Reduce  4  m.  5  fur.  13  rods,  to  feet. 

54.  Reduce  17  m.  6  fur.  20  rods,  8  feet,  to  inches. 

55.  Reduce  21£  17s.  3d.  to  pence. 

56.  Reduce  24£  to  sixpences. 

57.  Reduce  95£  3s.  to  sixpences. 

.  58.  Reduce  45£  5  s.  to  threepences. 

59.  Reduce  84£  to  fourpences. 

60.  In  1  cwt.  3  qrs.,  how  many  times  7  lbs.  ? 

61.  In  8  bis.  of  flour  at  7  qrs.  each,  how  many  parcels  of 
14  lb.  each  ? 

62.  In  13  bis.  of  cider,  at  31^  galls,  each,  how  many  timef 
8  gallons  ? 

63.  In.  a  town  5  miles  wide,  and  6  long,  how  many  acres  ? 

64.  How  many  acres  in  7500  sq.  miles  ? 


REDUCTION.  13^ 

SECTION    VI. 

KEDUCTION. 
Lower  denominations  are  reduced  to  higher  by  Division. 

Examples. 

1.  In  348  shillings  how  many  £  ? 

2.  Keduce  5000  shillings  to  £. 

3.  Reduce  13680  shillings  to  £. 

4.  Reduce  11040  pence  to  £. 

5.  Reduce  11292  pence  to  £. 

6.  Reduce  20220  pence  to  £. . 

7.  Reduce  1405  pence  to  £. 

8.  In  678  sixpences  how  many  £. 

9.  Reduce  549  threepences  to  £.  '' 

10.  Reduce  974  threepences  to  shillings. 

11.  Reduce  1776  hours  to  days. 

12.  Reduce  13841  hours  to  days. 

13.  Reduce  1964210  minutes  to  days. 

14.  Reduce  3742196  seconds  to  hours. 

15.  Reduce  964  gills  to  gallons. 

16.  Reduce  84672  gills  to  gallons. 

17.  Reduce  6794  gallons  of  wine  to  bis. 

18.  Reduce  3469  qts.  to  pecks. 

19.  Reduce  96431  pints  to  bushels. 

20.  Reduce  3846  qts.  to  bushels. 

21.  Reduce  5674  rods  to  furlongs. 

22.  Reduce  38961  rods  to  miles. 

23.  Reduce  76381  feet  to  rods. 

24.  Reduce  7960  inches  to  rods. 

25.  Reduce  7126734  inches  to  miles. 

26.  How  many  steps  of  2^  feet  each,  are  there  in  1  mile? 

27.  A  man  walks  30  miles  ;  how  many  steps  does  he  take, 
2|  feet  each  ? 

28.  Reduce  179  lb.  av.  to  cwt. 
,    29.  Reduce  413  lb.  to  cwt. 

30.  Reduce  1048  oz.  to  quarters. 

31.  Reduce  4352  drams  to  lb. 

32.  Reduce  6130  oz.  to  cwt. 

33.  Reduce  1280  dwt.,  Troy,  to  Ibs^ 


132  REDUCTION. 

34.  Reduce  1511  dwt.  to  lbs. 

35.  Reduce  17812  grs.  to  lbs. 

36.  Reduce  720  square  inches  to  square  feet. 

37.  Reduce  1029  square  feet  to  yards. 

38.  Reduce  2203  square  inches  to  yards. 

39.  Reduce  3267  square  feet  to  rods. 

40.  Reduce  5631  square  feet  to  rods.  _ 

41.  Reduce  86  solid  feet  to  yds. 

42.  Reduce  191934  solid  inches  to  feet. 

43.  Reduce  2333  solid  inches  to  feet. 

44.  Reduce  876  solid  feet  to  yds. 

45.  Reduce  2293  solid  inches  to  yds. 

46.  In  92  qrs.  cloth,  how  many  E.  E? 

47.  Reduce  361  nails  to  qrs. 

48.  Reduce  467  nails  to  yds. 

49.  Reduce  3,741  inches  to  E.  E. 

50.  Reduce  467  yds.  to  E.  E. 

51.  In  27  acres,  2  roods,  17  rods,  how  many  lots  of  32  rods 


52.  How  many  times  does  a  carriage  wheel  11^  feet  in 
circumference,  go  round  in  one  mile  ? 

53.  In   35    tons  weight,  how   many  wagon  loads   of   22 
cwt.  each  ? 

54.  In  186£  12s.,  how  many  guineas  of  21  shillings  each? 

55.  In  75  yds.  how  many  E.  E  ? 

56.  How  many  cannon  balls  at  24  lbs.  each,  will  it  take  to 
weigh  1  ton  gross  weight? 

57.  How  many  times  must  you  apply  a  pole  12  feet  long, 
to  the  ground,  to  measure  1  mile  ? 

58.  How  many  10  gallon  kegs  may  be  filled  from  17  hhd., 
wine  measure  ? 


SECTION    VII. 

COMPOUND  ADDITION. 

When  numbers  are  used  without  being  applied  to  any  par- 
ticular kind  of  quantity,  as  67,  84,  they  are  called  Abstract 
Numbers ;  when  they  are  applied  to  some  particular  quantity, 
as  67  yards,  they  are  called  Denominate  Numbers. 


COMPOUND   ADDITION. 


133 


When  several  numbers  of  different  denominations  are  to 
be  added,  as  3£  7s.4-7£  4s.,  it  is  called  Compound  Addition. 


Examples. 
1.  3£  14s.  9d+14£  lis.  6d. 


Operation. 

£  s.  d. 

3  14  9 

14  11  6 

18       6  3 


Ans. 


Set  down  numbers  of  the  same  denomination  under  each 
other.  Add  first  the  numbers  of  the  lowest  denomination ; 
if  the  sum  amounts  to  more  than  one  of  the  next  higher,  set 
down  what  is  over,  and  carry  the  number  of  the  higher  to 
the  next  column.  So  proceed  through  the  whole ;  in  adding 
the  last  column  set  down  the  whole  amount. 

2.  5£  15s.  4d.+14£  17s.  lld.+2£  6s.  5d. 

3.  13£  14s.  6d.  2qr.+65£  17s.  lOd.  Iqr. 

4.  48£  16s.+73£  10s.+91£  15s.+16£  17s. 

5.  17s.  3d.+15s.  7d.  2qr.+14s.  Od.  2qr. 

6.  Troy  weight.  12  lbs.  1  oz.  16  dwt.  14  grs.-}-3  dwt.  17  grs. 

7.  17  lbs.  2  oz.  16  dwt.  5  gr.4-6  lbs.  14  dwt.  17  grs. 

8.  21  lbs.  0  oz.  0  dwt.  3  grs.-t-19  dwt.  19  grs.+13  dwt. 

9.  21  lbs.  3  oz.  16  dwt.  15  grs.4-4  lb.  4  oz.  17  dwt.  13  grs. 

10.  Av.  wt.  gross.  3  cwt.  0  qr.  17  lb.  14  oz.-}-12  cwt. 
8  qrs.  13  lbs.  12  oz. 

11.  1  T.  14  cwt.  3  qrs.  17  lb.+3  T.  17  cwt.  1  qrs.  21  lbs. 

12.  3  T.  16  cwt.  1  qr.  20  lbs.  6  oz.+5  cwt.  3  qrs.  19  lbs. 
13  oz. 

13.  14  cwt.  1  qr.  20  lbs.+18  cwt.  1  qr.  16  lbs.+17  cwt. 
1  qr.  11  lbs. 

14.  1  m.  3  fur.  17  r.  6  ft.+3  m.  5  fur.  36  r.  12  ft. 

15.  65  m.  7  fur.  31  r.+18  m.  19  fur.  23  r.+19  m.  4  fur. 


17  r. 

16. 
17. 
18. 
19. 
20. 
21. 
22. 


5  r.  15  ft.4-27  r.  14  ft.+16  r.  11  ft.+21  r.  12  ft. 

15  r.  9  ft.  6  in.+17  r.  3  ft.  4  in.+25  r.  15  ft.  11  in. 
3  fur.  17  r.  4  ft.  5  in.+5  fur.  16  r.  14  ft.  9  in. 

7  fur.  16  r.  3  ft.  2  in.4-6  fur.  34  r.  12  ft.  10  in. 

13  m.  7  fur.  31  r.+6  m.  3  fur.  22  r.-j-ll  m.  5  fur.  8  r. 

Square  measure.    2  A.  3  R.  6  p.+15  A.  1  R.  17  p. 

16  A.  2  R.  21  p.+8  A.  3  R.  33  p.+9  A.  2  R.  9  p. 

12 


134  COMPOUND    SUBTRACTION. 

23.  2  R.  15  p.  63  ft.  29  in.+l  R.  17  p.  31  in.4-37  p.  18 
inches. 

24.  13  p.  45  ft.  18  in.+19  p.  3  ft.  23  in.+17  p.  64  ft.  71 
inches. 

25.  Solid  measure,  3  yds.  17  ft.  126  in.+4  yds.  23  ft.  64  in. 

26.  19  yards,  3  feet,  61  inches4-2  yards,  26  feet,  1650 
inches-f-4  yards,  18  feet,  91  inches. 

27.  16  gals.  3  qts.  1  pt.+84  gals.  2  qts.  1  pt. 

28.  13  bushels,  3  pks.  4  qts.-f-76  bushels,  3  pks.  5  qts. 

29.  19  bu.  1  pk.+76  bu.  3  pks.+18  bu.  2  pks. 

30.  14  yds.  3  qrs.  1  n.+21  yds.  2  qrs.  3  n. 

31.  3  days,  16  hours,  23  minutes-[-17  days,  13  h.  51  m. 

32.  1  year,  11  weeks,  4  days-}-3  y.  14  w.  2  d. 

33.  7  deg.  14  min.  34  sec.+lS  deg.  20'  30". 

34.  21  deg.  1>  11"+14  deg.  18'  19". 

35.  61  deg.  26'  14"+34  deg.  1'  8". 


SECTION    VIII. 

COMPOUND    SUBTRACTION. 

Compound  Subtraction  is  the  subtraction  of  numbers 
of  different  denominations. 

Rule,  Set  numbers  of  the  same  denomination  under  each 
other.  Begin  at  the  right  hand,  setting  down  the  remainder 
found  by  subtraction,  under  its  own  denomination.  If,  in  any 
case,  the  minuend  is  less  than  the  subtrahend,  borrow  one 
from  the  next  higher  denomination  of  the  minuend. 

Examples. 

1.  15£  8s.  9d.— 11£  lis.  4d. 

2.  22£  19s.  8d.— 18£  15s.  9d. 

3.  13£  4s.  6d.— 9£  15s.  lOd. 

4.  18  bushels,  3  pecks,  4  quarts, — 16  bushels,  2  p^s.  5  qts. 

5.  44  bushels,  1  peck,  3  quarts, — 20  bushels,  2  pks.  6  qts. 

6.  4  years,  3  months,*  14  days, — 2  years,  4  months,  18  d. 

I  — —'t 

*  Allow  30  days  to  a  month. 


Operation. 

£ 

s. 

d. 

15 

8 

9 

11 

11 

4 

3   17  5An8. 


COMPOUND    MULTIPLICATION.  135 

7.  28  years,  8  months,  5  days, — 19  years,  11  months,  2  days. 

8.  18  gallons,  3  quarts,  1  pint, — 10  gallons,  1  quart,  1  pint. 
9v  14  gallons,  1  quart, — 2  quarts,  1  pint. 

10.  4  miles,  3  furlongs,  17  rods, — 3  miles,  4  furlongs,  21  rods. 

11.  19  miles,  7  furlongs,  11  rods, — 9  miles,  6  furlongs,  13  r. 

12.  5  cwt.  3  quarters,  14  pounds, — 4  cwt.  1  quarter,  20  lbs. 

13.  12  cwt.  2  qrs.  21  lb.,— 9  cwt.  3  qrs.  23  lbs. 

14.  The  battle  of  Bunker  Hill  was  on  June  17,  1775  ;  the 
battle  of  Long  Island,  August  27,  1776  ;  what  was  the  length 
of  time  between  them  ? 

15.  The  battle  of  the  Brandy  wine  was  September  11, 1777 ; 
how  long  was  that  after  the  battle  of  Long  Island  ? 

16.  The  battle  of  Monmouth  was  June  28,  1778  ;  how  long 
was  that  after  the  battle  of  the  Brandy  wine  ? 

17.  The  army  of  Burgoyne  was  captured  October  17, 1777 ; 
that  of  Cornwallis,  October  9,  1781 ;  how  long  between  these 
events  ? 

18.  If  I  give  a  note  on  interest,  June  5,.  1839,  and  pay  it 
March  10, 1841,  for  how  long  a  time  must  the  interest  be  cast  ? 

19.  If  I  give  a  note  on  interest,  August  17, 1841,  and  pay  it 
June  9,  1843,  for  what  time  must  the  interest  be  cast  ? 

20.  How  long  is  it  from  Dec.  17,  1843,  to  June  6,  1844  ? 

21.  How  long  from  September  9;  1842,  to  Aug.  3,  1844  ? 

22.  How  long  from  January  16,  1840,  to  July  17,  1843  ? 

23.  How  long  from  Nov.  14,  1841,  to  August  21,  1844? 

24.  Boston  is  in  longitude  71°  4'  W. ;  New  York,  74°  1' ; 
what  is  the  difference  of  longitude  ? 

25.  Cincinnati  is  in  longitude  84°  27' ;  how  many  degrees 
W.  from  Boston  ? 

26.  How  many  degrees  of  longitude  is  Cincinnati  west  from 
New  York  ? 

27.  How  many  degrees  of  longitude  is  Cincinnati  west  from 
Philadelphia,  whose  longitude  is  75°  11'? 


SECTION    IX. 

COMPOUND  MULTIPLICATION. 


Multiply,  first,   the  lowest  denomination ;   if  the  product 
amounts  to  more  than  one  of  the  next  higher,  set  down  what 


136 


COMPOUND   DIVISION. 


is  over,  and  carry  the  number  of  the  next  higher  to  the  next 
product.  Multiply  the  next  denomination  in  the  same  way, 
and  so  on. 

Examples. 


Opferatlon. 

T.  cwt.  qr.     ^ 
3      7  3 

7 

U  1  Ans. 


23 


1.  3  T.  7  cwt.  3  qr.  multiplied  by  7. 

2.  Multiply  4£  5s.  6d.  by  2. 

3.  6£4s.  3d.  lqr.X3. 

4.  6  hours,  43  min.  15  sec.X4:. 

5.  9h.  11m.  41  sec.X6. 

6.  14  days,  17  hours,  15  minutes,  3  seconds  X  8. 

7.  8  bushels,  3  pecks,  1  quart  X 16. 

8.  15  bushels,  2  pecks,  3  quarts,  1  pintXl2. 

9.  19  bushels,  1  peck,  2  quarts  X  5. 

10.  1  mile,  5  furlongs,  13  rods,  12  feetX2. 

11.  13  miles,  2  furlongs,  4  rods,  6  feet,  3  inches X  8. 

12.  2cwt..3qrs.  16  lbs. X 7. 

13.  14  cwt.  3  qrs.  14  lbs.X4. 

14.  What  is  the  weight  of  12  casks  of  lime,  each  weighing 
3  cwt.  1  qr.  17  lbs.? 

15.  How  many  yards  in  9  pieces  of  calico,  each  measuring 
23  yards,  3  qrs.  ? 

16.  Troy  weight.    6  lbs.  11  oz.  5  dwt.X7. 

17.  9  lbs.  8  oz.  16  dwt.  4  grs.X9. 

18.  Square  measure.    7  acres,  2  roods,  17  rods X. 9- 

19.  15  acres,  1  rood,  34  rodsXl4. 


SECTION    X, 


COMPOUND    DIVISION. 

Divide  the  highest  denomination  first,  and  set  the  quotient 
under  it;  reduce  the  remainder,  if  any,  to  the  next  lower 
denomination,  add  it  to  those  of  the  same  in  the  dividend,  and 
divide  again  ;  and  so  on  to  the  end. 

Examples. 

1.  Divide  7  bu.  3  pks.  5  qts.  by  2. 


Operation, 
bu.  pk.  qt. 
2)7     3      5 

3     3     ejAns. 


MISCELLANEOUS   EXAMPLES.  1S7 

2.  3£  lis.  6d.-r-2. 

3.  18£  12s.  9d.^3. 

4;  26  hours,  53  miiautes-H4. 

5.  55  hours,  10  minutes,  6  seconds-r-6. 

6.  114  days,  18  hours,  0  minutes,  24  seconds-^8. 

7.  140  bushels,  2  pecks-Hl6. 

8.  187  bushels,  1  peck,  2  quartsH-12. 

9.  Seven  men  are  entitled  to  equal  shares  of  67£  13s.  4d. ; 
what  is  each  man's  share  ? 

10.  Three  men  are  to  receive  equal  shares  of  114£  19s.  9d. ; 
what  is  each  man's  share  ? 

11.  What  is  1  fourth  of  13  lbs.  6  oz.  17  dwt.,  Troy? 

12.  A  teamster  has  7  T.  11  cwt.  3  qrs.  of  merchandise, 
which  he  loads  on  three  wagons,  giving  an  equal  load  to 
each  ;  how  much  was  each  load  ? 

13.  If  you  divide  7  bushels  and  3  pecks  of  oats  equally 
among  5  horses,  how  much  will  each  receive  ? 

14.  If  a  piece  of  land  containing  35  acres,  3  roods,  14  rods, 
be  divided  into  4  equal  parts,  how  much  will  each  part  be  ? 


SECTION    XI. 

MISCELLANEOUS  EXAMPLES. 

1.  A  teamster  loads  a  quantity  of  merchandise  equally  on 
8  wagons,  putting  on  each  1  T.  11  cwt.  2  qrs. ;  finding  these 
loads  too  heavy,  he  takes  a  fourth  wagon  ;  how  much  must  he 
load  on  each,  to  divide  the  whole  equally  among  the  four  ? 

2.  A  man's  estate  amounts  to  784£  10s. ;  his  wife  is  to 
receive  214£  15s.  and  the  remainder  is  to  be  divided  equally 
among  4  children  ;  what  will  be  each  child's  share  ? 

3.  Three  men  have  equal  shares  in  a  scaffold  of  hay,  the 
whole  of  which  weighs  5  T.  11  cwt. ;  what  is  each  man'^  share  ? 

For  the  following  examples  see  p.  40,  Part  I. 

4.  Rome  is  in  longitude  12°  28'  E.  from  London ;  what 
time  is  it  at  Rome  when  it  is  noon  in  London  ? 

5.  Petersburg  is  in  longitude  29°  48'  E. ;  what  time  is  it  at 
Petersburg  when  it  is  noon  in  London  ? 

12* 


138  DIVISIBILITY   OF  NUMBERS. 

6.  Paris  is  in  longitude  2°  20'  E. ;  what  time  is  it  at  Paris 
when  it  is  noon  in  London  ? 

7.  Boston  is  in  longitude  71°  4'  "W. ;  what  time  is  it  in 
Boston  when  it  is  noon  at  London  ? 

8.  New  York  is  in  longitude  74°  1/  W. ;  what  time  is  it  in 
New  York  when  it  is  noon  at  London  ? 

9.  What  time  is  it  in  Cincinnati,  84°  27'  W.,  when  it  is 
noon  in  Boston,  which  is  71°  4'  W.  ? 


SECTION    XII. 

DIVISIBILITY  OF  NUMBERS. 

In  order  to  ascertain  if  a  number  is  divisible  by  either  of 
the  following  numbers,  2,  3,  4,  5,  6,  8,  9,  10,  or  any  combina- 
tion of  these,  see  Sec.  VIII,  Part  I. 

To  ascertain  if  a  number  is  divisible  by  any  other  number 
than  the  above,  make  trial  of  other  prime  divisors,  as  7,  11, 
13,  17,  &c.,  beginning  with  the  smallest,  tiU  you  find  one  that 
will  divide  the  given  number,  or  find  that  it  is  indivisible. 

Remember,  that  in  making  trial  by  these  numbers,  you 
need  not  go  higher  than  the  square  root  of  the  given  number, 
for  if  a  number  is  divisible,  one  of  the  factors  will  certainly 
be  as  small  as  the  square  root.  Let  us  take  the  number  1079 ; 
what  are  its  prime  factors  ?  By  inspection  you  may  see  it  is 
not  divisible  by  2,  3,  5,  or  11,  consequently  not  by  4,  6,  8,  9, 
10,  or  12.  On  trying  it  by  7,  it  is  found  not  divisible  by  7 ;  the 
next  number  is  13 ;  this  divides  it,  giving  a  quotient,  83,  which 
is  prime.     Its  only  factors,  therefore,  are  13  and  83. 

Examples. 

1.  What  are  the  prime  factors  of  667  ? 

2.  What  are  the  prime  factors  of  406  ? 

3.  What  are  the  prime  factors  of  419  ?  of  361  ?  of  742  r 
of  281?   of  316? 

4.  Prime  factors  of  941  ?  812?  749?  1116?  246?  8104. 

5.  Prime  factors  of  266?  884?  1917?  376? 


REDUCTION   OF  FRACTIONS.  139 


SECTION    XIII. 

REDUCTION   OF  FKACTIONS. 

[See  Section  VIII.  Part  I.] 

1.  Reduce  f  f  to  its  lowest  terrfis.     Ans.  ^. 

2.  Reduce  |§  to  its  lowest  terms. 

3.  Reduce  -^x^  to  its  lowest  terms. 

4.  Reduce  ff  to  its  lowest  terms. 

5.  Reduce  ^V^  to  its  lowest  terms. 

6.  Reduce  ^|^  to  its  lowest  terms.  In  this  example  it  is 
not  evident  on  inspection  whether  the  two  terms  of  the  frac- 
tion have  any  common  divisor.  In  such  cases  you  may  adopt 
the  following  Rule  to  find 

27ie   Greatest  Oommon  Divisor. 

Divide  the  greater  number  by  the  less,  and  then  take  the 
divisor  for  a  new  dividend,  and  divide  it  by  the  remainder, 
and  so  on,  till  there  is  no  remainder ;  the  last  divisor  will  be 
the  greatest  common  divisor. 

Apply  the  above  rule  to  the  sixth  example. 

187)22]_(1  ' 

187 

~o7\1Qj/K  The   greatest   common  divisor  is, 

,    -1 70  therefore,  17,  and,  dividing  the  terms 

of  the  fraction  by  this,  we  have  for 

17)34(2  the  lowest  terms,  U. 
34  ^^ 

00 

Demonstration  of  the  Rule. 

If  the  larger  number  is  a  multiple  of  the  smaller,  it  is  evi- 
dent that  the  smaller  is  a  common  divisor  of  the  two  numbers ; 
it  is  also  the  gi'eatest  common  divisor ;  for  a  number  cannot 
be  divided  by  any  number  greater  than  itself;  the  answer, 
therefore,  is  found  by  the  first  division.  But  if  there  is  a 
remainder,  next  find  whether  the  remainder  will  exactly 
divide  the  divisor.  If  it  will,  it  will  divide  both  the  original 
numbers,  for  if  it  will  divide  the  divisor,  it  will  divide  any 
multiple  of  the  divisor  ;  and,  as  it  will  of  course  divide  itself, 
it  will  divide  any  multiple  of  the  divisor,  plus  itself.    Now  the 


\ 

140  THE    GREATEST   COMMON   DIVISOR. 

larger  of  the  original  numbers  is  a  certain  multiple  of  tLe 
smaller,  plus  the  remainder.  If,  therefore,  after  the  first 
division,  the  remainder  will  divide  the  divisor,  it  is  a  common 
divisor,  or  measure,  of  the  two  numbers. 

It  is  also  the  greatest  common  divisor ;  for,  as^  it  will  ex- 
actly measure  the  smaller  of  the  two  numbers,  it\vill  exactly 
measure  any  multiple  of  the  smaller.  Now  the  greater  num- 
ber, is  a  certain  multiple  of  the  smaller,  plus  the  remainder. 
The  remainder,  therefore,  in  measuring  the  larger  number,  is 
obliged  to  measure  itself.  No  number  greater  than  itself  can 
do  this  ;  therefore  the  remainder  is  the  greatest  common  divi- 
sor. If  the  work  has  to  be  carried  on  farther  than  the  second 
division,  the  same  reasoning  in  the  demonstration  will  apply. 

JUxamples. 

7.  What  is  the  greatest  common  divisor  of  874  and  437  ? 

8.  What  is  the  greatest  common  divisor  of  497  and  451  ? 

9.  What  is  the  greatest  common  divisor  of  817  and  913  ? 

10.  What  is  the  greatest  common  divisor  of  1007  and 
1219? 

11.  What  is  the  greatest  common  divisor  of  608  and  192  ? 

12.  What  is  the  greatest  common  divisor  of  869  and  1343  ? 

When  there  are  more  than  two  numbers,  first  find  the 
greatest  common  divisor  of  two  of  them,  and  then,  of  that 
divisor,  and  the  third  number. 

13.  What  is  the  greatest  common  divisor  of  608,  941  and 
451? 

Whenever  it  is  possible,  by  inspection,  to  separate  the 
numbers  into  their  prime  factors,  this  method  should  be 
adopted. 

14.  What  is  the  greatest  common  divisor  of  94,  804  and 
126? 

15.  What  is  the  greatest  common  divisor  of  1274,  896  and 
580? 

Apply  the  above  Rules  to  the  reduction  of  the  following 
fractions. 

16.  Reduce  ^|-f-  to  its  lowest  terms. 

17.  Reduce  to  their  lowest  terms  f §^,  ^ff,  ^Hf. 

18.  Reduce  to  their  lowest  terms  |^|,  ^f  |,  §|i. 
19-.  Reduce  to  thciis^lowest  terms  t^^,  ^^,  1^. 


CHANGE    OF   NUMBERS   TO   HIGHER   TERMS.  141 

To  reduce  an  improper  fraction  to  a  whole  or  mixed  number. 

Perform  the  division  indicated  by  the  fraction  as  far  as 
possible ;  if  there  is  a  .remainder,  express  that  part  of  the 
division  by  placing  the  denominator  under  the  remainder. 

20.  Reduce  \^  to  a  whole,  or  mixed  number.     Ans.  l-^^. 

21.  Reduce  ^-  to  a  whole,  or  mixed  number.     Ans.  3|. 

22.  Reduce  to  a  whole,  or  mixed  number,  3^,  f f,  ff . 

23.  Reduce  to  a  whole,  or  mixed  number,  ^,  ^|,    y^., 

24.  Reduce  the  improper  fractions,  Vif  j  ^f  ^>  Wj   ^%^' 


SECTION    XIV. 

CHANGE  OF  NUMBERS  AND  FRACTIONS  TO  HIGHER  TERMS. 

It  is  sometimes  convenient  to  express  whole  numbers  in  the 
form  of  fractions,  and  to  express  fractions  in  higher  terms 
without   altering  the  value.     Thus   3=|,  or    ^.     10=^, 

or  V*-  ' 

Examples. 

1.  In  4  how  many  fifths?     Ans.  20. 

2.  Express  the  value  of  4  m  fifths.     Ans.  ^. 

3.  Express  7  in  thirds. 

4.  Express  19  in  the  form  of  sevenths. 

5.  In  13  how  many  eighths  ? 

6.  Express  21  in  thirds. 

7.  Express  7  in  eighteenths. 

8.  Express  41  in  fourths. 

9.  In  3^  how  many  halves  ? 

10.  Change  4^  to  an  improper  fraction. 

11.  Change  17^  to  an  improper  fraction. 

12.  Change  24^  to  an  improper  fraction. 

13.  Change  to  an  improper  fraction  18|.    112f.    318^. 

14.  Change  f  to  eighths,  without  altering  its  value. 

15.  Change  f  to  fifteenths. 

16.  Change  I  to  24th3.    ^3^  to  80ths.    f  to  99ths. 

17.  Change  /^  to  26ths.     ^  to  56ths.     -ff  to  60ths. 

18.  Change  |  to  7ths. 

This  example  presents  a  difficulty,  because  the  required 
denominator,  7,  is  not,  as  in  the  preceding  examples,  a  multi- 


142       MULTIPLICATION   AND    DIVISION    OP   FRACTIONS. 

pie  of  the  given  denominator,  4.  We  have  seen,  however, 
that  if  we  multiply  or  divide  both  terms  of  a  fraction  by  the 
same  number,  the  value  will  not  be  altered.  We  must  then 
multiply  and  divide  both  terms  by  such  numbers  as  will  give 
us,  in  the  end,  7  for  the  denominator.  The  question  then  is, 
how  can  we,  by  multiplication  and  division,  change  4  into  7  ? 
We  can  multiply  it  by  7,  which  will  give  28,  and  then  divide 
by  4,  giving  7  for  the  quotient.  Thus  the  denominator  has 
been  changed,  by  multiplication  and  division,  from  4  to  7. 
Now  whatever  has  been  done  to  the  denominator  must  be  done 
to  the  numerator  to  preserve  the  value  of  the  fraction.  Multi- 
plying 3  by  7,  we  have  21 ;  dividing  this  by  4  we  have  5^  for 

the  required  numerator.     The  answer,  therefore,  is  -3.     This 

fraction,  as  one  of  its  terms  contains  a  fraction  in  itself,  is 
called  a  complex  fraction. 

19.  Change  f  to  8ths.    |  to  9ths.    ^j  to  7th3. 

20.  In  f ,  how  many  4ths  ?  how  many  5ths  ?  6ths  ? 

21.  Change  4|  to  5ths.    8^  to  llths.    7^- to  4ths. 

22.  Change  22^  to  4ths.  18^  to  7ths.  31^  to  5ths. 

23.  In  8^,  how  many  3ds  ?  4ths?  5ths?  9ths? 

24.  In  19  J  how  many  5ths  ?  4ths?  7ths? 

25.  In  9^  how  many  3ds?  5ths  ?  8ths  ? 

26.  In  13^  how  many  14ths  ?  15ths? 

27.  In  8^  how  many  17ths?   13ths? 

28.  In  20^  how  many  7ths  ?  8ths  ? 

29.  In  16^  how  many  4ths  ?  5ths  ? 

30.  In  ll|  how  many  37ths  ?  19ths  ? 


SECTION    XV. 

MULTIPLICATION  AND  DIVISION  OF  FRACTIONS. 

[Sec  Section  VIII.  Part  I.] 

1.  A  man  worked  72  days  for  f  of  a  dollar  a  day ;  what 
did  his  wages  amount  to  ? 

2.  Multiply  I  by  46. 

3.  A  man  bought  139  bush,  of  apples  for  |  of  a  dollar  a 
bush. ;  what  did  they  come  to  ? 

4.  Multiply  f  by  341.    ^X127. 


MULTIPLICATION   AND   DIVISION   OF   FRACTIONS.       143 

5.  A  garrison  of  700  soldiers  are  allowed  f  of  a  pound  of 
flour  a  day,  for  each  man  ;  how  much  would  they  consume  in 
1  day  ?     How  much  in  7  days  ? 

6.  If  a  horse  eats  -^^  of  a  bush,  of  oats  in  a  day,  how  many 
bush,  will  he  eat  in  365  days? 

7.  If  a  horse  eats  2^  cwt.  of  hay  in  a  week,  what  part  of  a 
cwt.  will  he  eat  in  one  day  ?  How  many  cwt.  will  he  eat  in 
a  year  ? 

8.  Three  men  gain  by  an  adventure  56^  dollars,  which 
they  are  to  share  equally ;  what  is  each  man's  share  ? 

9.  What  is  ^  of  187^  ?     What  is  ^  of  91f  ? 

10.  f^X  141?  AX 97?  1X140? 

11.  If  3:^  cwt.  of  flour  be  divided  equally  into  5  equal  parts, 
what  part  of  a  cwt.  will  each  share  be  ? 

12.  Divide  17^  by  8.    37^  by  14.    18|  by  9. 

13.  Divide  74^-T-5.     811^-7.     381^-^-9. 


SECTION    XVI. 

MULTIPLICATION  AND  DIVISION  OF  FRACTIONS. 

[See  Section  IX.  Part  I.] 

1.  In  63  gallons  how  many  bottles  of  j^  of  a  gal.  each  ? 

2.  From  7  lbs.  of  flour  how  many  loaves  of  bread  may  be 
made,  each  containing  §  of  a  lb.  of  flour  ? 

3.  Divide  f f^f .    il^.    {^-i-U- 

4.  A  man  left  5846  dollars ;  f  of  the  whole  to  go  to  his  wife, 
and  the  remainder  to  be  equally  divided  among  four  children; 
what  was  each  child*s  share  ? 

5.  From  a  piece  of  cloth  32  yds.  long,  how  many  coats  can 
be  made,  each  requiring  2|  yds.  ? 

6.  From  a  stick  of  timber  2Q^  feet  long,  how  many  blocks 
can  be  cut,  each  -^^  of  a  foot  long  ? 

7.  If  a  family  consume  22^  lbs.  of  flour-  in  a  week,  how 
much  is  that  a  day  ?     How  much  will  they  use  in  a  year  ? 

8.  Divide  13i-H^.    18f-r-lU.    86^9^. 

9    Multiply  25^X16^.    23|X2f.    31fXl9^. 
A  fraction  of  a  fraction,  as  ^  of  ^,  is  called  a  Compound 
Fraction.     This  is  reduced  to  a  simple  fraction  by  multiply- 


144        ADDITION   AND    SUBTRACTION    OF   FRACl?IONS. 

ing  the  numerators  together  for  a  new  numerator,  and  the  de- 
nominators for  a  new  denominator.  It  is  in  fact  the  same  as 
the  multiplication  of  two  fractions  together. 

10.  What  is  i  off  of  76? 

11.  What  is  ^  of  ^  of  12  ?   What  is  |  of  f  of  18^  ? 

12.  What  is  ^^  of  f  of  8J  ?     What  is  f  of  |  of  34  ? 

13.  Divide  f  of  |  by  ^.     Divide  f  of  f  by  8^. 

14.  Multiply  ^  of  37  by  19^.     Mult.  ^  of  18  by  19^. 

15.  What  is  the  value  of  32:^  yds.  of  cloth  at  4^  dollars  a 
yard? 

16.  What  do  17|^  tons  of  hay  come  to  at  11|^  dolls,  a  ton? 

17.  What  is  the  value  of  21^  cords  of  wood  at  4^  dolls,  a 
cord? 

18.  What  is  the  value  of  24^  ban-els  of  apples  at  If  dolls, 
a  barrel  ? 

19.  What  is  the  amount  of  12|^  shares  Bank  stock,  at  641iJ- 
dollars  a  share  ? 


SECTION    XVII. 

ADDITION  AND   SUBTRACTION  OF  FRACTIONS. 

If  the  fractions  have  a  common  denominator,  perform  the 
required  operation  on  the  numerators,  and  place  the  result 
over  the  common  denominator. 

1.  Add  A+A+l-    tV+i^+if    1^+^+1%- 

2.  Subtract  f  —  |.     ^— ^.     «  — H-     M  — H- 

If  the  fractions  have  not  a  common  denominator,  reduce 
them  to  a  common  denominator,  (Sec.  IX.  Pt.  1.)  and  then 
add  or  subtract,  as  the  question  requires. 

s.  Add  m+f .  f +A+if .  T^F+t+f . 

4.  Add  f +1+1^,     i+A+i§.     I+A+A- 

5.  Subtract  5i^  —  f     S^\  —  i'     |  — iV 

6.  Add  3U+57i+18i.     19^+211+3/^. 

In  cases  like  the  above,  it  is  easiest  to  add  the  whole  num- 
bers first. 

7.  Add  ei^+l-  of  7^+^  of  22.     4f\+^  of  18^+|  of  1 3. 

8.  A  man  spent  for  various  articles  ^  of  a  dollar,  f  of  a 
dollar,  ^  o£  &  dollar,  ^  of  a  dollar ;  what  part  of  a  dollar 
did  he  spend  in  all  ? 


REDUCTION    OP   DENOMINATE   FRACTIONS.  145 

9.  From  56^  bushels,  llf  bushels  were  taken  ;  how  much 
remained  ? 

10.  From  a  ^firkin*  of  butter  containing  4^2^  lbs.,  18];^  lbs. 
were  taken ;  how  much  remained  ? 

11.  Add  ^+4|.       1U+I|.       15i+7f+g. 

12.  Add  2H+^.     n+^1,     13^+^+11^. 

13.  Add  19I+21H.     8i+H.     26i+H+g. 

14.  Add4f+^.     8^+^,     12^+3^+18^. 

15.  Subtract  19i—^.     22^  — ?|.     18^  —  13/^. 

16  4d 

16.  Add  3i+8f+Hi_LA.     4U4ii-fi^. 


SECTION    XVIII. 

BEDUCTION  OF  DENOMINATE  FRACTIONS. 

Denominate  fractions  are  fractions  of  numbers  when  ap- 
plied to  a  particular  denomination.     See  Sec.  XI.  Part  I. 

Mcamples. 

1.  What  part  of  a  bushel  is  ^  of  a  quart?     f  of  a  quart? 

2.  What  part  of  2  bushels  is  ^  of  a  quart  ?     f  of  a  peck  ? 

3.  What  part  of  7  bushels  is  f  of  a  peck  ?     |  of  a  peck  ? 

4.  What  part  of  3£  is  ^  of  a  shilling  ?     y^  of  a  shilling  ? 

5.  What  part  of  a  shilling  is  |  of  a  penny  ?     ^^3^  of  a  d.  ? 

6.  What  part  of  a  mile  is  ^  a  rod  ?      ^  of  a  rod  ? 

7.  What  part  of  3  furlongs  is  |  of  a  rod  ?     f  of  a  rod  ? 

8.  What  part  of  a  mile  is  3  furlongs  19  rods? 

9.  What  part  of  a  mile  is  1  foot  ?     2  feet  ? 

10.  What  part  of  a  ton  is  6  cwt.  3  qrs.  7  lbs.  ? 

11.  What  part  of  a  ton  is  18  cwt.  1  qr.  6  lbs.  ? 

12.  What  part  of  a  square  rod  is  17  feet?     31 1  feet? 

13.  What  part  of  a  cord  is  12^  cubic  feet?     25^  cubic  feet? 

14.  What  part  of  a  cord  is  18;^  cubi«  f«et  ?     84^  €ubi«  f«et  ? 

18 


146  CHANGE    OF   DENOMINATE    INTEGERS. 

15.  What  part  of  a  bushel  is  ^  of  a  quart  ?     f  of  a  qt.  ? 

16.  What  part  of  a  bushel  is  7^  qts.  ?     11^  qts.  ? 

17.  What  part  of  a  week  is  4|^  hours  ?     5^  hours  ? 

18.  What  part  of  a  week  is  7^  hours  ?     8^  hours  ? 

19.  What  part  of  3  hours  is  12  minutes?     15^  minutes? 

20.  What  is  the  value  in  shillings  and  pence  of  ^  of  a  £  ? 
If  it  were  3£,  there  would  be  60  shillings ;  but  it  is  not  3£, 

but  one  seventh  of  that ;  therefore,  it  is  }  of  60  shillings=8f 
shillings.  To  find  the  value  in  pence  of  ^  of  a  shilling,  pursue - 
the  same  reasoning ;  if  it  was  4  shillings,  it  would  be  48  pence ; 
but  it  is  not  4  shillings,  but  ^  of  thatz=:6f  pence.  The  answer, 
then,  is  8s.  6f  d. 

21.  What  is  the  value  in,  shillings  and  pence  of  f  of  a  £? 

22.  Value  in  hours,  minutes,  and  seconds,  of  |  of  a  week  ? 

23.  How  many  minutes  in  ^i^  of  a  week  ? 

24.  How  many  minutes  and  seconds  in  ^  of  an  hour  ? 

25.  Value  of  f  of  a  gal.  ?     Value  of  A  ^^  a  bl.  of  wine  ? 

26.  How  many  oz.,  dwt.  and  grs.in  /^  of  a  lb.  Troy? 

27.  What  is  the  value  in  oz.,  dwt.  and  grs.  of  ^j  of  a  lb.  Troy  ? 

28.  How  many  square  rods  in  f  of  an  acre  ? 

29.  How  many  square  feet  and  inches  in  ^  of  a  sq.  yard  ? 


SECTION    XIX. 

CHANGE  OF  DENOMINATE  INTEGERS  TO  FRACTIONS. 

[See  Section  XI.  Part  I.J 

Examples. 

1.  What  part  of  a  furlong  is  5  feet  ?     1^  feet  ? 

2.  What  part  of  a  mile  is  17  feet  ?     28:^  feet  ? 

3.  What  part  of  a  mile-ts  31^  feet ?     47§  feet? 

4.  What  part  of  a  ton  is  25^  lbs.  ?     82|  lbs.  ? 

5.  What  part  of  a  ton  is  107^  lbs  ?     130^  lbs. 

6.  What  part  of  a  ton  is  3  qrs.  9  lbs.  8  oz.  ?     17^  lbs. 

7.  What  part  of  a  ton  is  4  lbs.  13  oz.  ?     19^^  lbs. 

8.  What  part  of  an  acre  is  4  rods  17  feet?     139^^  feet? 

9.  What  part  of  an  acre  is  113  rods  51^  feet? 

10.  What  part  of  a  hhd.  of  wine  is  17  gals.  3  qts.  1  pt.  ? 

11.  What  part  of  10  gallons  is  3^  pints?     8^  pints? 

12.  What  part  of  a  guinea  is  3s.  7d.  ?     14s.  3^  ? 


PEACTIONS.  147 

13.  What  part  of  a  £  is  8s.  5id.  ?     10s.  9f  d.  ? 

14.  What  part  of  a  week  is  3|  hours?     1^  hours? 

15.  What  part  of  5  days  is  1  hour,  41  m.  15  sec.  ? 

16.  What  part  of  a  month  of  31  days,  is  17  h.  18^  m.? 

17.  What  part  of  an  ell  Eng.  is  2  n.  1  in.  ? 

18.  What  part  of  21  yards  is  3^  qrs.?     9^. yards? 

19.  What  part  of  37^  yards  is  1^  ell  English  ?     - 

20.  What  part  of  7  cords  of  wood  is  21^  cubic  feet? 

21.  What  part  of  31£  is  5s.  S^d.  ?     14s.  2|d.  ? 


SECTION    XX. 

PKACTICAL  EXAMPLES. 

I.  What  is  the  cost  of  If  yds.  broadcloth  at  4|^  dolls,  a  yd.  ? 
^^  2.  What  is  the  cost  of  2\  yds.  cloth  at  ^  of  a  doll,  a  yd.  ? 

3.  What  is  the  value  of  12  bbls.  of  flour  at  4|  dolls,  a  bbl.  ? 

4.  What  is  the  value  of  31  casks  of  lime  at  ^  of  a  dollar  a 
cask? 

5.  What  is  5^  cwt.  of  beef  worth  at  4f  dolls,  per  cwt.  ? 

6.  What  is  the  cost  of  43^  bush,  corn  at  f  of  a  doU.  a  bush.  ? 

7.  If  a  horse  eat  1^  bush,  of  oats  in  a  week,  how  much  will 
he  eat  in  52  weeks  ? 

8.  What  will  be  the  cost  of  the  oats  for  52  weeks  at  f  of  a 
doll,  a  bushel  ? 

9.  What  part  of  a  cwt.  is  ^  of  a  bbl.  of  flour  containing 
196  lbs.? 

10.  What  is  the  weight  of  f  of  a  lot  of  hay  weighing  4^ 
tons? 

II.  How  many  cubic  feet  are  there  in  f  of  J  of  a  cord  of 
wood? 

12.  A  man  sold  f  of  a  lot  of  wood,  the  whole  of  which  was 
17f  cords ;  how  much  did  he  sell  ? 

13.  What  part  of  9  rods  in  length  is  10  feet  ? 

14.  What  part  of  a  sq.  rod  is  3  sq.  yards  ? 

15.  What  part  of  a  sq.  rod  is  5  sq.  feet  ? 

16.  What  is  36  sq.  feet  of  land  worth  at  9^  dolls,  a  sq.  rod  ? 

17.  How  many  gallons  are  there  is  \^  of  a  barrel  of  wine  ? 

18.  Two  men  bought  a  lot  of  hay  for  11^  dollars  ;  one  took 
13  cwt. ;  the  other,  the  remainder,  which  was  8^  cwt. ;  what 
•?ught  each  to  pay  ?         _ 


148  DECIMAL    FRACTIONS. 

19.  Two  men  divided  a  lot  of  wood  wliicli  they  purchased 
together  for  27|^  dollars;  one  took  5^  cords;  the  other  8 
cords  ;  what  ought  each  to  pay  ? 

20.  The  main-spring  of  a  watch  weighs  about  1  dwt.  12 
grs.,  Troy  weight ;  estimating  its  worth  at  ^  of  a  dollar,  what 
would  a  pound  Troy  of  steel  be  worth  after  it  was  manufac- 
tured into  watch  main-springs,  allowing  nothing  for  waste  in 
manufacturing  ? 

21.  A  hair-spring  of  a  watch  weighs  |  of  a  grain,  Troy; 
estimating  its  value  at  3  cents,  what  would  be  the  value  of  1 
lb.  Troy,  of  steel,  made  into  hair-springs,  allowing  nothing  for 
waste  ? 

22.  Two  men  hired  a  horse  one  week  for  6^  dollars  ;  one 
rode  him  70  miles ;  the  other,  84 ;  how  much  ought  each  to 
pay? 

23.  A  stack  of  hay  is  bought  by  two  men  for  76^  dollars, 
to  be  paid  for  in  proportion  to  the  amount  of  hay  each  one 
takes ;  one  takes  3|  tons,  the  other  the  remainder,  which  was 
2^  tons ;  how  much  ought  each  to  pay? 


SECTION    XXI. 

DECIMAL    F.RACTIONS. 
Addition  and  Subtraction.     Sec.  XII.  Pt.  I. 

Examples. 

1.  24.5+68.3+17.14+87.96+3.125. 

2.  165.3+96.45+8.431+.641+9412.5. 

3.  450.61+27.134+89.4216+984. 

4.  64.25+3.125+87.25+181.7. 

5.  125.17+34.27+.125+3761.5. 

6.  186.4—27.31;   800.4—21.67. 

7.  34.21—18.525;    94.31—81.167. 

8.  167.51—35.125;    204.5—31.09. 

9.  20.41—3.817;    601.4—517.24. 

10.  648.62— .541;   346.4—91.324. 

11.  5.1—1.324;   .5— .0067. 

12.  .81— .126;   .94— .3816. 


EEDUCTION    OF    FRACTIONS    TO    DECIMALS.  149 

Multiplication  and  Division. 

13.  1243X87;   321.67X24.3.  31.  35H-.36;  48-f-.47. 

14.  97.125X6;  31.4X.125.      "  32.  .17-^-31;  .26-^.013. 

15.  37.5X.94;    18.4X64.  33.  43-H.06 ;  45-^.003. 

16.  21X-106;  312X.05.  34.  75H-.125 ;  .95-i-.04. 

17.  31.1X.004;  18.61X.03.  35.  .18-i-.0045 ;  11H-.34. 

18.  641X.41;  843.5X.95.  36.  9-^.0225;  .7-^-.035. 

19.  184.2X.121;   35.6X.025.  37.  80-i-.18 ;  51-V-.031. 

20.  .625X71;   .875X31.5.  88.  .55X.031;  71.4X.ia. 

21.  84H-.012;  965-f-.15.  39.  8.44-.021 ;  .65H-.8. 

22.  1.65-H15;  846^3.4.  40.  1.21X.09  ;' .14X.03. 

23.  1640^.96;  425H-.055.  41.  .64X.31;  .08X.009. 

24.  l-T-.OOl;  2-^.0002.  42.  36-J-.13 ;  28-^-11.4. 

25.  .OOl-f-2;  384-^-.0012.  43.  40.1-f-8;  64-^.9. 

26.  96-^-.024;  64-H.016.  44.  81.4^.03;  7-f-.4. 

27.  1827^.9;  34-^.17.  45.  9-^.5;  15H-.7. 

28.  .63-^-8;  .15-H-14.  46.  80.2X.03  ;  16-r-.9. 

29.  .48^.9;   .33—16.  47.  105.4-f-37.15. 

30.  181-H-.41;  41-H.6.  48.  118.75~.0044. 


SECTION    XXII. 

EEDUCTION  OF  VULGAB  FRACTIONS  TO  DECIMALS. 
[See  Section  XITI.  Part  1. 3 

Examples. 

1.  Reduce  f  to  a  decimal. 

2.  Reduce  f  to  a  decimal. 

3.  Reduce  to  decimals  \  ;  f . 

4.  Reduce  to  decimals  iV ;  -^  5  i^e* 

5.  Reduce  to  decimals  t^  ;  i^ ;  M* 

If  the  fractions  are  reducible  to  decimals  without  a  remainder, 
obtain  the  answer  exactly ;  if  they  are  irreducible,  obtain  the 
proximate  answer  to  four  places,  and  annex  the  fractional 
remainder.  In  order  to  know  if  a  fraction  is  exactly  expres- 
sible in  decimals,  see  Section  XIII.  Part  I,  as  directed  above. 

6.  Reduce  to  decimals  ^ ;   §| ;   §|-. 

7.  Reduce  to  decimals  ^ ;  ^t  5  ^* 

8.  Reduce  to  decimals  \^ ;  -^ ;  jf  5. 

13* 


150  REDUCTION    OF   FRACTIONS    TO    DECIMALS. 

9.  Reduce  to  decimals  :^  ;  ^|f  ;  |^. 

10.  Reduce  to  decimals  y^ij  ;  y^^  ;  ^f  x* 

In  ordinary  transactions  it  is  usual  to  carry  the  decimal 
answer  to  three  or  four  places ;  the  remainder  is  then  so 
email  in  value  that  it  may  be  dropped  as  of  no  importance. 
At  whatever  place  you  stop,  however,  the  decimal  obtained, 
and  the  fractional  remainder,  when  added  together,  will  ex- 
actly equal  the  original  fraction. 

11.  In  order  to  show  this,  we  will  take  \.     Reducing  it, 

')_       we  obtain  at  the  first  step  1  tenth-M-  of  1  tenth; 

1+^ 
adding  these,  7^+^x7==^^=^'  which  is  the  original  faction. 

We  now  carry  the  reduction  one  step  further,    ^ — -' 

14-|-^ 

we  obtain  14  hundredths,-f-f  of  a  hundredth.     Adding  these, 

^+Y%7f=^%=h  the  original  faction. 

We  will  carry  the  reduction  one  step  further ;     I 

^  ^  142+f. 

We  obtain  142  thousandths+f  of  a  thousandth.  Adding  these 
by  using  the  common  denominator  7000,  TVA+Tif(T^=^§^^ 
s=:^,  the  original  fraction. 

12.  Reduce  -^  to  a  decimal,  of  one  figure,  with  the  remain- 
der ;  carried  to  2  places,  with  the  remainder ;  carried  to  3 
places,  with  the  remainder. 

13.  Reduce  -j^  to  a  decimal  of  7  places. 

14.  Reduce  ^^  to  a  decimal  of  9  places. 

15.  Reduce  ^  to  a  decimal  of  10  places. 

Repeating  and  Oirculating  Decimals. 

When  a  fraction  is  irreducible,  the  decimal  figure  will  either 
repeat,  as  ^=.333-[- ;  or  the  decimal  figures  obtained  by  the 
partial  reduction  will,  after  a  time,  recur  again,  in  the  same 
order  as  at  first.  Thus,  ^  gives  .090909-}-  and  so  on,  with- 
out end.  When  the  same  figure  is  repeated  continually,  it  is 
called  a  repeating  decimal ;  when  the  same  series  of  different 
figures  recurs,  it  is  called  a  circulating  decimal. 


▲NALTSIS  OF  PBOBLEMS.  15t 

SECTION    XXIII. 

BEDUCTION    OF    DENOMINATE    INTEGERS    TO    DECIMALS. 

1.  Reduce  5s.  lid.  to  the  decimal  of  a  £.  First,  reduce 
the  quantity  to  the  vulgar  fraction  of  a  £ ;  then  reduce  that 
vulgar  fraction  to  a  decimal. 

2.  Reduce  Ss.  2^.  to  the  decimal  of  a  £. 

3.  Reduce  5d.  to  the  decimal  of  a  guinea. 

4.  Reduce  3  qts.  to  the  decimal  of  a  bushel. 

5.  Reduce  2^  pints  to  the  decimal  of  a  gallon. 

6.  Reduce  3  feet  5  inches  to  the  decimal  of  a  rod, 

7.  Reduce  7  feet  8  inches  to  the  decimal  of  a  rod. 

8.  Reduce  15  rods  9^  feet  to  the  decimal  of  a  furlong. 

9.  Reduce  23  rods  13  feet  to  the  decimal  of  a  mile. 

10.  Reduce  5  hours  18  m.  to  the  decimal  of  a  day. 

11.  Reduce  21  hours  6  m.  to  the  decimal  of  a  week. 

12.  Reduce  12^  sq.  rods  to  the  decimal  of  an  acre. 


SECTION    XXIV. 

TO  FIND  THE  INTEGRAL  VALUE  OF  DENOMINATE  DECIMALS. 

1.  Wliat  is  the  value  of  .7  of  a  rod  ? 

Supposing  the  quantity  was  7  rods,  its  value  in  feet  would 
be  found  by  multiplying  it  by  16^  ;  16^X7=115^,  or  115.5 ; 
but  it  was  not  7  rods,  but  7  tenths,  of  a  rod,  whose  value  we 
wish  to  find  ;  the  answer  obtained,  therefore,  is  10  times  too 
large;  dividing  by  10,  it  is  11.55, — 11  feet  and  55  hundredths. 
In  order  to  find  the  value  in  inches  of  55  hundredths  of  a 
foot,  we  will  call  it  55  feet;  the  answer  is,  55X12=660  — 
660  feet ;  but,  as  we  regarded  the  55  as  100  times  greater 
in  value  than  it  is,  the  answer  is  100  times  too  large ;  divid- 
ing it  by  100,  the  answer  is  6.60  inches,  =  6  inches  and  60 
hundredths,  or  6  tenths. 

The  above  analysis  shows  the  nature  of  the  operation  in 
all  cases. 


152  PRACTICAL    EXAMPLE^. 

'  2.  What  is  the  value,  in  feet  and  inches,  of  .3  of  a  rod  ? 

3.  What  is  the  value  of  .94  of  a  rod  ? 

4.  What  is  the  value  of  .26  of  a  rod  ? 

5.  How  many  shillings  and  pence  are  there  in  .65  of  a  £? 

6.  How  many  shillings  and  pence  are  there  in  .8  of  a  £  ? 

7.  How  many  pence  are  there  in  .7  of  a  shilling  ? 

8.  How  many  pence  are  there  in  .16  of  a  shilling? 

9.  What  is  the  value  of  .19  of  a  £? 

10.  What  is  the  value  of  .74  of  a  bushel  ? 

11.  What  is  the  value  of  .9  of  a  bushel? 

12.  What  is  the  value,  in  rods  and  feet,  of  .7  of  an  acre  ? 

13.  What  is  the  value  of  .9  of  an  acre  ? 

14.  What  is  the  value  of  .12  of  an  hour? 

15.  How  many  minutes  and  seconds  in  .15  of  an  hour? 

16.  Find  the  value  of  .34  of  a  week. 

17.  Find  the  value  of  .162  of  a  week. 

18.  Find  the  value  of  .84  of  a  minute. 

19.  How  many  feet  in  .761  of  a  cord? 

20.  How  many  feet  and  inches  in  .2  of  cord  ? 

21.  How  many  feet  in  .74  of  a  cord  ? 

22.  How  many  feet  in  .13  of  a  cord? 


SECTION    XXV. 

PRACTICAL   EXAMPLES. 

1.  Add  $1.50+$.3754-$.0625+$.1875+$5.00. 

2.  Add  $34.75-[-^6.00-4-$.3754-$.08. 

3.  A  man  had  S50,  and  spent  $.375  of  it ;  how  much  had 
he  left? 

4.  A  man  had  $10.00,  and  spent  $.875  of  it ;  how  much  had 
he  left? 

5.  A  watch  cost  $45.675 ;  the  chain  and  key,  $4.845 ;  what 
did  the  whole  cost  ? 

6.  The  owner  then  sold  the  watch,  chain,  and  key,  for 
$48,375  ;  how  much  did  he  lose? 

7.  A  man  set  out  on  a  journey  with  $10.00;  the  first  day 
he  spent  $1,125  ;  how  much  had  he  left? 


DECIMAL    FRACTIONS.  158 

8.  The  second  day  he  spent  $1.425 ;  how  much  had  he  left  ? 

9.  The  third  day  he  spent  S1.67 ;  how  much  h^d  he  left  ? 

10.  The  fourth  day  he  spent  $.875;  how  much  had  he 
left? 

11.  What  is  the  cost  of  21  lbs.  of  flour  at  $.05  per  lb.? 
Why  do  you  point  off  two  decimals  in  the  answer  ? 

12.  What  is  the  cost  of  35  lbs.  of  flour  at  $.045  per  lb.? 
Why  do  you  point  off  three  decimals  ? 

13.  What  is  the  cost  of  12.5  lbs.  of  flour  at  $.05  a  lb.  ? 

14.  What  is  the  cost  of  15.5  lbs.  of  flour  at  $.045  a  lb.? 

15.  What  is  the  cost  of  26.25  lbs.  of  flour  at  $.0375  a  lb.? 

16.  What  is  the  cost  of  13.75  lbs.  of  flour  at  $.0425  a  lb.  ? 

17.  What  is  the  cost  of  15  barrels  of  flour  at  $4.75  a  bar- 
rel? 

18.  What  is  the  cost  of  17.5  barrels  of  flour  at  $5.25  a 
barrel  ? 

19.  What  is  the  cost  of  3  tons  of  hay  at  $7.56  a  ton  ? 

20.  What  is  the  cost  of  13.5  tons  of  hay  at  $9.00  a  ton  ? 

21.  What  are  17  barrels  of  cider  worth  at  $1.75  a  barrel? 

22.  What  cost  16  gallons  of  molasses  at  $.345  a  gallon? 

23.  Divide  $1.05  into  21  equal  parts;  what  will  each  part 
be? 

24.  How  many  lbs.  of  flour  will  $15.75  buy,  at  $.045  a  lb.  ? 

25.  How  many  times  is  $.05  contained  in  $.625  ? 

26.  How  many  lbs.  of  flour  can  be  bought  for  $.6975,  at 
$.045  per  lb.  ? 

27.  How  many  times  is  $.0375  contained  in  $984,375  ? 

28.  How  many  times  is  $.0425  contained  in  $584,375  ? 

29.  How  many  barrels  of  flour  will  $71.25  buy,  at  $4.75 
per  barrel  ? 

30.  How  many  barrels  of  flour  will  $91,875  buy,  at  $5.25 
per  barrel  ? 

31.  How  many  tons  of  hay  can  be  bought  for  $22.68,  at 
$7.56  per  ton? 

32.  How  many  times  is  $9.00  contained  in  $121.50  ? 

33.  A  shipmaster  paid  $29.75  for  ballast,  giving  $1.75  a 
ton  ;  how  many  tons  did  he  buy  ? 

34.  How  many  times  is  $.345  contained  in  $5.52  ? 

35.  What  cost  14  lbs.  of  flour  at  $.045  a  lb.,  .and  28  lbs. 
of  sugar  at  $.095  a  lb.  ? 


154  FRACTIONS. 


SECTION    XXVI. 

PRACTICAL    QUESTIONS    IN    VULGAR    AND    DECIMAL 
FRACTIONS. 

1.  Bought  7  cwt.  15  lbs.  sugar  at  S6.62|  per  cwt.,  and  sold 
it  at  7  cents  per  lb.     What  was  the  gain  ? 

2.  Bought  156  gallons  of  wine  at  93  cents  per  gallon,  and 
sold  it  at  34  cents  per  quart.     What  was  the  gain  ? 

3.  Bought  7  cwt.  1  qr.  11  lbs.  coffee  at  $12.50  per  cwt.,  and 
sold  it  at  14  cents  per  lb.     What  gain  ? 

4.  Bought  37  yards  broadcloth  at  $5.25  per  yard ;  sold  20 
yards  of  it  at  S7.00  per  yard,  and  the  remainder  at  $6.31  per 
yard.     What  was  the  gain  ? 

5.  Bought  24  yards  broadcloth  at  $6.40  per  yard;  sold  22f 
yards  at  $7.25  per  yard,  and  the  remnant  for  5  dollars.  What 
was  the  gain  ? 

6.  Bought  87  E.  E.  calico  at  17  cents  per  E.  E.,  and  sold 
it  at  21  cents  per  yard.     What  gain  ? 

7.  Bought  4  dozen  books  at  $1.50  per  dozen,  and  sold 
them  at  16  cents  each.     What  gain? 

8.  Bought  13  dozen  brooms  at  $1.04  per  dozen,  and  sold 
them  at  15  cents  each.     What  gain? 

9.  Bought  5^  dozen  mats  at  $3.40  per  dozen,  and  sold  them 
at  36  cents  each.     What  gain  ? 

10.  Bought  17  bushels  of  salt  at  65  cents  per  bushel,  and 
sold  it  at  21  cents  per  peck.     What  gain  ? 

11.  Bought  one  barrel  of  wine  at  78  cents  per  gallon,  and 
sold  it  at  16  cents  per  pint.     What  gain? 

12.  Bought  3  dozen  baskets,  at  $2.05  per  dozen,  and  sold 
1  dozen  at  31  cents ;  1  dozen  at  37  cents ;  and  1  dozen  at  42 
cents  each.     What  gain  ? 

13.  Bought  48  yards  broadcloth  at  $5.62  per  yard ;  lost 
17  yards  by  fire,  and  sold  the  remainder  at  $6.25  per  yard. 
How  much  gain  or  loss  ? 

14.  Bought  a  hhd.  molasses  containing  131  gallons,  at  34 
cents  per  gallon  ;  1 6  gallons  leaked  out ;  sold  the  remainder 
at  37  cents  per  gallon.     What  gain  or  loss  ? 

15.  Bought  3^  dozen  axes  at  $6.80  per  dozen,  and  sold 
them  at  92  cents  each.     What  gain  ? 

16.  Bought  7  dozen  pails  at  $1.42  per  dozen,  and  sold  them 
at  21  cents  each.     What  gain  ? 


REDUCTION    OF   CURRENCIES.  155 

17.  Bought  8J  dozen  shovels  at  S9.25  per  dozen,  and  sold 
thera  at  $1.00  each.     What  gain  ? 

18.  Bought  74  yards  carpeting  at  73  cents  per  yard,  and 
sold  it  at  87^  cents  pei*  yard.     What  gain? 

19.  Bought  164  bushels  corn  at  54  cents  per  bushel;  sold 
93  bushels  at  67  cents,  and  the  remainder  at  50  cents  per 
bushel.     How  much  loss  or  gain  ? 

20.  Bought  75  barrels  apples  at  $1.37  per  barrel ;  lost  15 
barrels  by  decay,  and  sold  what  remained  at  $  2.12  per  bar- 
rel.    What  loss  or  gain  ? 

21.  Bought  13  dozen  oranges  at  7  cents  per  dozen ;  lost  by 
decay  2^  dozen,  and  sold  the  remainder  at  2^  cents  each. 
What  gain  ? 

22.  Bought  15  dozen  pairs  of  shoes  at  $4.87  per  dozen, 
and  sold  them  at  63  cents  per  pair.     What  gain  ? 

23.  Bought  18^  thousand  of  boards  at  $9.50  per  thousand; 
sold  6  thousand  at  $12.25  per  thousand,  and  the  remainder  at 
$8.42  per  thousand.     What  gain  ?. 

24.  Bought  21^  cords  wood  at  $4.75  per  cord ;  sold  8  cords 
at  $5.50  per  cord,  and  the  remainder  at  $4.25  per  cord. 
What  gain  or  loss  ? 

25.  Bought  209  bushels  apples  at  27  cents  per  bushel ;  sold 
46  bushels  at  49  cents  per  bushel,  and  the  remainder  at  25 
cents  per  bushel.     What  gain  ? 


SECTION    XXVII. 

REDUCTION    OF    CUEEENCIES. 
English   Currency. 
1.  Reduce  67£  to  dollars  and  cents. 

As  4s.  6d.  or  54d.=$1.00  (see  Table,  p.  35,)  and  20s.  or 
240d.=l£,  1  dollar  is  t^-^q  of  a  £.  Reducing  this  fraction  to 
its  lowest  terms,  it  is  ^^.  The  question  therefore  is  this  ;  in 
67£  how  many  ^  of  a  £  ?  Dividing  67  by  the  fraction,  we 
have  297^  dollars,  for  the  answer.  The  fraction  ^  gives  77 
cents  and  7  mills. 


156  REDUCTION  OP  CURRENCIES. 

2.  Reduce  87£  to  dollars  and  cents. 

3.  Reduce  104£  to  dollars  and  cents. 

4.  Reduce  64£  to  dollars  and  cents. 

5.  Reduce  167£  to  dollars  and  cents. 

6.  Reduce  520£  to  dollars  and  cents. 

7.  Reduce  84£  6s.  to  dollars  and  cents. 

First  reduce  the  6s.  to  the  decimal  of  a  £,  ^j=.3 ;  the 
sum  then  is,  84.3£.  Reduce  it  in  the  same  way  as  the  cases 
above. 

8.  Reduce  124£  13s.  to  Federal  money. 

9.  Reduce  36£  9s.  6d.  to  Federal  money. 
10.  Reduce  71£  18s.  4d.  to  Federal  money. 

To  Reduce  Federal  money  to  Sterling. 

11.  In  684  dollars,  how  many  £,  s.  and  d.  ? 

As  $1.00=^^  of  a  £,  1£=^  of  $1.00.  The  question 
therefore  is,  in  684  dollars,  how  many  ^  of  Sl.OO  ?  Divid- 
ing by  the  fraction  we  have  for  the  answer,  £153.9,  or  153£ 
18s. 

12.  In  $74.25,  how  many  £,  s.  and  d.? 

13.  Reduce  $186.40  to  Sterling  money. 

14.  Reduce  $564.35  to  Sterling  money. 

15.  Reduce  $640.15  to  Sterling  nioney. 

The  comparative  value  of  the  dollar  and  the  pound  sterling, 
as  given  above,  is  called  the  nominal  par  value.  The  actual 
value  of  the  £,  is  higher  than  is  here  given.  This  difference 
is  usually  estimated  in  trade  by  adopting  the  nominal  par 
value,  given  above,  as  the  basis  of  the  calculation,  and  then 
adding  or  subtracting  a  certain  per  cent,  as  8  or  10  per  cent., 
to  compensate  for  the  inequality  of  value. 

Canada  Currency. 
5s.=60d.=$1.00. 

16.  In  74£  15s.,  how  many  dollars  and  cents. 

As  $1.00=60d.,  and  l£=240d,  $1.00  is  ^,  or  ^  of  a  £  ; 
multiplying  by  4,  the  answer  is  $299.00. 

1 7.  In  126£  12s.  Canada  currency,  how  many  dollars  and 
cents  ? 

18. .  Reduce  $841.50  to  Canada  currency. 


INTEREST.  157 

New  England  Currency. 
6s.=72d.=$1.00. 

19.  In  64£  8s.  how  many  dollars  and  cents  ? 
$1.00=3^,=^^^  of  a  £.     Eeduce  the  8s.  to  a  decimal  of 

a  £,  and  divide  by  the  fraction ;  we  have  $214.66f . 

20.  Reduce  120£  12s.  6d.  to  Federal  money. 

New   York  Currency. 
8s.=96d.==$1.00. 

21.  Reduce  146£  6s.  4:d.  to  Federal  money. 

'  As  $1.00=T^j=-^  of  a  £,  reducing  the  shillings  and 
pence  to  the  decimal  of  a  £,  and  dividing  by  the  fraction,  we 
have  $365.75. 

22.  Reduce  54£  10s.  6d.  to  Federal  money. 

PennsylvaniaT  Currency. 
7s.  6d.=90d.=S1.00. 

23.  Reduce  16£  5s.  6d.  to  Federal  money. 
$1.00=/^=!  of  a  £. 

24.  Reduce  7£  8s.  9d.  to  Federal  money. 


SECTION    XXVIII. 

INTEREST. 
[See  Section  XTV.  Part  I.J- 

Ruh.  Find  the  interest  of  1  dollar  for  the  given  time ; 
multiply  the  principal  by  it,  and  point  off  as  in  the  multipli- 
cation of  decimals. 

1.  What  is  the  interest  of  S156.34  for  11  months  and  20 
days? 


As  the  interest  of  1  dollar  for  2 
months  is  1  cent,  for  10  months,  it 
will  be  5  cents,  .05.  As  the  interest 
of  1  dollar  for  6  days  is  1  mill,  for 
30  days  it  will  be  5  mills,  and  for  20 
days,  3  mills  and  ^,  making  8  mills 
and  \.  Set  down  the  8  at  the  right 
14 


3)156.34 

.058 

125072 

78170 

5211 

$9.11983    Ans. 


158  INTEREST. 

hand  of  the  .05,  and  for  the  ^  divide  by  3.  Observe,  the  0 
before  the  5  must  be  retained,  otherwise  it  would  be  5  tenths 
of  a  dollar,  or  50  cents,  and  the  answer  would  be  10  times 
too  great.  If  there  are  no  cents,  there  must  be  two  ciphers 
at  the  left  hand  of  the  mills.  The  number  of  cents  for  the 
multiplier  is  always  equal  to  half  the  greatest  even  number 
of  months,  —  the  number  of  mills  is  one  sixth  of  all  the  days 
over  and  above  the  greatest  even  number  of  months. 

2.  Interest  of  $384.18  for  7  months  and  10  days  ? 

3.  Interest  of  $147.19  for  5  months  15  days? 

4.  Interest  of  $568.25  for  9  months  13  days? 

5.  Interest  of  $81.40  for  10  months  14  days  ? 

6.  Interest  of  $56.32  for  12  months  24  days? 

7.  Interest  of  $75.30  for  14  months  18  days  ? 

8.  Interest  of  $644.46  for  15  months  24  days? 

9.  Interest  of  $831.00  for  1  year  4  months  12  days  ? 

10.  Interest  of  $380.00  for  1  year  7  months  ? 

11.  Interest  of  $500.00  for  1  year  5  months  6  days? 

12.  Interest  of  $27.42  for  4  months  17  days  ? 

13.  Interest  of  $13.18  for  6  months  23  days  ? 

14.  Interest  of  $1000.00  for  5  months  4  days  ? 

15.  Interest  of  $65.48  for  30  days,  or  1  month? 

16.  Interest  of  $94.00  for  30  days  ? 

17.  Interest  of  $840.60  for  18  days  ? 

18.  Interest  of  $632.00  for  18  days? 

19.  Interest  of  $349.40  for  12  days  ? 

20.  Interest  of  $267.62  for  12  days  ? 

21.  Interest  of  $384.92  for  15  days? 

22.  Interest  of  $811.19  for  20  days? 

23.  Interest  of  $673.94  for  5  months  11  days  ? 

24.  Interest  of  $460.00  for  8  months  18  days  ? 

25.  Interest  of  $460.00  for  8  months  18   days,  at  12  per 
cent.  ? 

26.  Interest  of  $460.00  for  8  months  18  days,  at  8  per 
cent.  ? 

27.  Interest  of  $460.00  for  8  months  18  days,  ^t  7  per 
cent.  ? 

28.  Interest  of  $460.00  for  8  months  18  days,  at  5  per 
cent.  ? 

29.  Interest  of  $1500.00  for  15  months,  at  4  per  cent.  ? 

30.  Interest  of  $145.80  for  7  months  11  days,  at  8  per 
cent.  ? 


INTEREST.  159 

31.  Interest  of  $341.18  for  2  years  9  months  IS  days? 
Asjhe  interest  of  a  dollar  for  30   days,  is  5  mills,  for  ^  of 

30  days,  or  b  days,  it  is  one  mill.   As  1  mill  is  txAtttj  ^^^  thou 
sandth  of  a  dollar,  it  follows  that  the  interest  of  1   dollar  for 
1  day,  is  one  sixth  of  a  thousandth,  or  ^^Vtt  ^^  ^  dollar.     For 
two  days,  therefore,  it  wiU  be  ^^^^j^ ;  for  15  days,  s^^j^  of  a 
dollar. 

A  convenient  rule,  therefore,  when  the  time  is  short,  is  the 
following  : 

Mvltiply  the  sum  hy  the  number  of  days,  and  divide  the 
product  hy  6000. 

This  is  often  the  shortest  method.  You  divide  by  1,000, 
by  removing  the  decimal  point  three  places  to  the  left.  It 
only  remains  then,  after  doing  this,  to  multiply  by  the  num- 
ber of  days,  and  divide  by  6. 

32.  What  is  the  interest  of  $348.25  for  18  days? 
Dividing  by  1000,  you  have  $0,348:^,  —  thirty  four  cents, 

eight  mills  and  a  quarter.     Instead  now  of  multiplying  by  18 
and  dividing  by  6,  you  may  multiply  by  3,  for  18  is  3  times  6. 

3  times  0.348^  is  $1.044|  —  Ans. 

33.  Interest  of  $725.80  for  24  days  ? 

34.  Interest  of  $341.18  for  36  days? 

35.  Interest  of  $67.45  for  54  days  ? 

36.  Interest  of  $641.18  for  42  days? 

37.  Interest  of  $84.16  for  15  days? 

To  find  the  amount,  add  the  interest  to  the  principal ;  or, 
find  the  amount  of  $1.00  for  the  given  time,  and  multiply 
the  principal  by  .it. 

38.  What  is  the  amount  of  $560.50  for  8  months  12  days? 

39.  Amount  of  $964.25  for  15  months  18  days? 

40.  Amount  of  $460.00  for  1  year  6  months  ? 

41.  Amount  of  $120.50  for  2  yeai-s  4  months? 

42.  Amount  of  $68.40  for  1  year  6  months  24  days? 

43.  Amount  of  $500.00  for  2  years  3  months  ? 

44.  Amount  of  $730.50  for  6  months  12  days? 

45.  Amount  of  $840.25  for  4  months  18  days  ? 

46.  Amount  of  $40.50  for  8  months  12  days  ? 


160  PARTIAL   PAYMENTS. 

SECTION    XXIX, 

PARTIAL    PAYMENTS. 


\ 


When  Partial  Payments  are  made  on  a  note,  the  amount 
due  on  the  final  payment  of  the  note,  may  be  found  by  the 
following  rule. 

Find  the  interest  on  the  note  up  to  the  time  of  the  first 
payment ;  if  the  payment  exceeds  the  interest,  deduct  it  from 
the  amount,  regarding  the  remainder  as  a  new  principal ;  on 
this,  calculate  the  interest  to  the  time  of  the  next  payment, 
and  so  on.  If  any  payment  is  less  than  the  interest  then  due, 
reserve  it,  and  compute  the  interest  on  to  the  time  when  the 
payments,  added  together,  shall  exceed  the  interest  due ;  then 
subtract  the  sum  of  the  payments  from  the  amount  then  due, 
and  proceed  as  before. 

1.  A  note  of  200  dollars  is  given  July  1,  1834,  on  which 
are  the  following  partial  payments ; 

Dec.  15,  1834,  $25.00. 
March  1,  1835,  2.50. 
Aug.  10,  1835,   45.00. 

What  was  due  Dec.  31,  1835  ? 

2.  A  note  of  $340.25  is  given  Aug.  1,  1840. 

Endorsements,  — Jan.  10,  1841,  $28.40. 
July  1,  1841,  9.00. 
March  14,  1842,  74.00. 

What  was  due  Jan.  1,  1843  ? 
8.  A  note  of  $480.00  is  given  June  9,  1841. 
Endorsements, —  Sept.  11,  1842,     $60.00. 

Jan.  3,  1843,  95.00. 

March  12,  1844,  100.00. 
What  was  due  Dec.  1,  1844? 
4.  A  note  of  $675.40  is  given  July  3,  1843. 
Endorsements, — Jan.  4,  1844,   $65.00. 

April  17,  1844,  29.50. 

Nov.  18,  1844,  74.00. 
What  is  due  Jan.  1,1845? 


PARTIAL    PAYMENTS. — ANNUAL   INTEREST.  161 

5.  A  note  of  $345.40  is  given  Aprin,  1843. 

Endorsements,  — Dec.  1,  1843,  S40.00. 

.  :  .June  10,  1844,  90.00. 

f  ^  Oct.  4,  1844,      31.50. 

Feb.  6,  1845,     17.00. 

What  is  due  Jan.  1,  1846  ?  ^ 

The  rule  given  above  is  the  legal  rule.  When,  ho;i^ver, 
the  note  is  paid  within  a  year  from  the  time  when  it  was 
given,  the  following  rule  is  usually  employed. 

Find  the  amount  of  principal  and  interest  of  the  whole 
note,  from  the  time  it  was  given,  till  the  final  payment. 

Find  the  amount  of  each  payment,  from  the  time  it  was 
paid,  till  the  final  payment ;  and  the  sum  of  these  amounts 
subtract  from  the  amount  of  the  whole  note ;  the  remainder 
will  be  the  balance  due. 

6.  A  note  of  $525.00  is  given  Sept.  1,  1844. 

Endorsements,  —  Dec.  30,  1844,    $58.75. 
March  4,  1845,  104.20. 
June  8,  1845,        63.40. 
What  is  due  Aug.  21,  1845  ? 

7.  A  note  of  $784.50,  given  July  7,  1844,  has  the  follow- 
ing endorsements, — 

Sept.  5,  1844,  $54.00. 

Nov.  10,  1844,  60.00. 

Jan.  12,  1845,  75.00. 

March  17,  1845,  100.00. 
What  is  due  May  1,  1845  ? 

ANNUAL   INTEREST. 

When  a  note  is  given,  payable  at  a  longer  period  than  a 
year  from  the  date,  it  is  usual  to  express  in  the  note,  that  the 
interest  shall  be  paid  annually.  At  the  end  of  a  year,  the 
holder  of  the  note  may  compel  the  payment  of  the  interest. 
In  such  cases  the  debtor,  instead  of  paying  the  interest  that 
is  due,  sometimes  renews  the  note,  adding  the  interest  to  the 
principal.  Thus,  at  the  end  of  each  year  the  interest  due 
is  added  in,  and  goes  to  make  a  new  principal  for  the  follow- 
ing year.  This  is  called  Compound  Interest ;  but  the  com- 
14* 


162  ANNUAL   INTEREST. 

putation  of  it,  is  the  same  as  in  simple  interest ;  for,  if  the 
interest  is  not  computed  every  year,  and  either  paid  or  put 
into  the  note  by  renewal,  that  interest  cannot  draw  interest.* 
The  law  regards  it  the  duty  of  the  creditor  to  remind  tj^e 
debtor  of  his  debt,  by  exacting  the  payment  of  the  interest^ 
every  year.  If  he  does  not  do  this,  he  can  derive  no  ad- 
vantage from  the  promise  in  the  note  to  pay  the  interest 
annually. 

ILLUSTRATION. 

Boston,  March  1,  1845. 


$100.00. 


For  value  received,  I  promise  to  pay  to  John  Jones,  or 
order,  one  hundred  dollars,  in  five  years,  with  interest  an- 

^"^1^-  Samuel  Barton. 

If  John  Jones  does  not  exact  the  interest  till  the  end 
of  the  five  years,  and,  if  he  obtains  no  renewal  of  it,  the 
amount  of  the  note  will  be  only  $130.00 ;  for  the  interest  of 
100  dollars,  for  five  years,  is  30  dollars. 

If,  however,  he  obtains  a  renewal  of  the  note  at  the  end 
of  each  year,  the  principal  of  the  note,  for  the  second  year, 
will  be  $106.00. 

8.  What  will  the  principal  of  the  note  for  the  third  year 
be? 

9.  What  will  the  principal  of  the  note  for  the  fourth  year 
be? 

10.  What  will  the  principal  of  the  note  for  the  fifth  year 
be? 

11.  What  will  be  due,  principal  and  interest,  at  the  end  of 
the  fifth  year  ? 

12.  How  much  would  the  holder  of  the  above  note  lose  by 
omitting  to  obtain  any  renewal  of  it,  or  any  payment  of  an- 
nual interest  ? 


$250.00  ITew  York,  July  1,  1845. 

For  value  received,  I  promise  to  pay  John  Foss  or  order, 
two  hundred  and  fifty  dollars,  in  four  years,  with  interest 
annually.  Amos  Carr. 

•  In  some  of  the  States,  the  interest,  alter  it  falls  due,  draws  simple  interest  tiU  it  is  paid. 


DISCOtTNT.  163 

13.  If  no  interest  is  paid  on  this  note  till  the  principal  is 
due,  and  if  no  renewal  of  the  note  is  made,  what  will  be  the 
amount  of  the  note  at  the  time  of  payment  ? 

14.  If  the  note  is  renewed  each  year,  what 'will  be  the 
principal  of  the  note,  for  the  second  year  ? 

15.  What  will  be  the  principal  of  the  note  for  the  third 
year? 

16.  What  will  be  the  principal  of  the  note  for  the  fourth 
year? 

17.  What  will  be  the  amount  of  the  last  note  at  the  time 
of  payment? 

18.  How  much  would  the  holder,  John  Foss,  lose,  by  neg- 
lecting to  obtain  any  annual  payment  of  interest,  or  any  re- 
newal of  the  above  note  ? 


SECTION    XXX. 

DISCOUNT. 
CSee  Section  XIV.  Part  I.] 

Examples. 

1.  What  is  the  present  worth  of  $475.50,  payable  in  3 
months  ? 

2.  What  is  the  present  worth  of  $341.00,  payable  in  65 
days? 

3.  Present  worth  of  $940.25,  payable  in  4  months  ? 

4.  Present  worth  of  $156.30,  payable  in  96  days? 

5.  Present  worth  of  $312.60,  payable  in  35  days? 

6.  Present  worth  of  $500.00,  payable  in  41  days  ? 

7.  Present  worth  of  $814.67,  payable  in  65  days  ? 

8.  Present  worth  of  $46.30,  payable  in  20  days  ? 

9.  Present  worth  of  $124.45,  payable  in  5  months  ? 
10.  Present  worth  of  $360.20,  payable  in  4^  months  ? 


164  BANKING.  ^ 

SECTION    XXXI. 

BANKING. 

[See  Section  XIV-  Part  I.] 

To  find  the  present  worth  of  a  note  given  to  a  bank,  pay- 
able at  some  future  time,  find  the  present  worth  of  1  dollar 
for  the  given  time,  and  multiply  the  sum  named  in  the  note 
by  it. 

1.  What  is  the  present  worth  of  a  note  for  100  dollars,  dis- 
counted at  a  bank,  for  60  days  ? 

Interest  of  1  dollar  for  63  days  is  .0105 ;  this  subtracted 
from  1  dollar,  leaves  for  the  present  worth  .9895. 

2.  What  is  the  present  worth  of  a  note  for  $450.00,  dis- 
counted at  a  Bank,  for  90  days  ? 

3.  I  give  my  note  to  a  bank  for  $250.00,  for  60  days ; 
what  do  I  receive  ? 

4.  I  give  my  note  to  a  bank  for  $520.00,  for  120  days ; 
what  do  I  receive  ? 

5.  Present  worth  of  a  bank  note  for  $600.00,  discounted 
for  60  days  ? 

6.  Present  worth  of  a  bank  note  for  $150.00,  discounted 
for  120  days  ? 

7.  Present  worth  of  a  bank  note  for  $75,00,  discounted  for 
30  days  ? 

8.  Present  worth  of  a  bank  note  for  $1000.00,  discounted 
for  60  days  ? 

9.  Present  worth  of  a  bank  note  for  $560,00,  discounted 
for  120  days? 

10.  Present  worth  of  a  bank  note  for  $150.00,  discounted 
for  30  days  ? 

To  find  what  must  be  the  face  of  a  note  given  to  a  bank, 
in  order  to  obtain  a  certain  sum  — find  the  present  worth  of  1 
dollar  for  the  given  time,  and  divide  the  sum  you  wish  to  obtain 
hy  it ;  the  quotient, will  express  the  sum  that  must  he  named  in 
the  note.  This,  you  observe,  is  just  the  reverse  of  the  pre- 
ceding case. 

11.  For  what  sum  must  I  give  my  note  to  a  bank,  payable 
in  60  days,  in  order  to  receive  $98.95  ? 


LOSS    AND    GAIN.  —  PER    CENTAGE.  165 

12.  For  what  sum  must  I  give  my  note  to  a  bank,  payable 
in  120  days,  in  order  to  receive  $509.34  ? 

13.  For  what  sum  must  I  give  my  note  to  a  bank,  payable 
in  60  days,  in  order  to  receive  $593.70  ? 

14.  For  what  sum  must  I  give  my  note  to  a  bank,  payable 
in  30  days,  in  order  to  receive  $10000.00  ? 


SECTION    XXXII. 

LOSS    AND   GAIN.  — PER    CENTAGE. 
[See  Section  XIV.  Part  I.] 

1.  A  man  bought  a  horse  for  75  dollars,  and  sold  him  for 
$82.50  ;  what  did  he  gain  per  cent.  ? 

2.  A  man  bought  a  chaise  for  $178.00,  and  sold  it  for 
$154.50  ;  what  did  he  lose  per  cent.  ? 

3.  A  merchant  bought  a  lot  of  flour  at  $4.62  a  barrel,  and 
sold  it  at  $5.15  a  barrel ;  what  was  his  gain  per  cent.  ? 

4.  A  merchant  bought  a  piece  of  broadcloth  for  $4.30  per 
yard ;  what  must  he  sell  it  for  to  gain  12  per  cent.  ? 

5.  A  man  has  $1200.00  invested  in  a  manufactory;  he 
receives,  for  his  half-yearly  dividend,  30  dollars ;  what  per 
cent,  is  that  on  his  stock  ? 

6.  A  merchant  fails,  owing  $8540.00,  and  can  pay  but 
$2700.00  ;  how  much  will  that  be  on  a  dollar  ? 

7.  A  man  failing  in  business,  agrees  to  pay  his  creditors 
87  cents  on  a  dollar ;  what  must  a  creditor  receive,  whose 
claim  is  $740.30  ? 

The  pupil  should  be  encouraged  habitually  to  reason  upon 
the  operations  he  performs  ;  so  that  his  method  of  procedure 
may  be  suggested  by  the  relations  of  the  numbers,  and  not 
dictated  by  a  special  rule.  To  aid  in  this  important  habit,  a 
few  remarks  will  be  made  on  some  of  the  foregoing  examples. 
These  may  serve  as  specimens  of  analysis,  and  suggest  to  the 
student  a  similar  course  of  reasoning  in  other  cases. 

Example  1.  The  whole  gain  is  $7.50  ;  if  this  gain  were 
made  on  an  outlay  of  one  dollar,  the  gain  would  be  •  seven 
hundred  and  fifty  per  cent. ;  but  the  gain  is  made  on  an  outlay 
of  75  dollars ;  the  gain  per  cent.,  therefore,  is  one  seventy-fifth 
of  the  whole  gain. 


166  PER    CE^TTAGfe. 

Example  4.  If  the  cost  was  1  dollar  a  yard,  lie  must  add 
12  cents ;  if  2  dollars,  he  must  add  24  cents,  &c. 

Example  5.  If  30  dollars  had  been  the  gain  upon  1  dollar, 
it  would  have  been  30  hundred  per  cent. ;  but  the  gain  was 
upon  1200  dollars ;  the  per  cent.,  therefore,  must  be  one 
twelve  hundredth  of  30  dollars. 

8.  I  invest  in  a  factory  1260  dollars^  and  receive  for  my 
yearly  dividend  86  dollars ;  what  is  that  per  cent.  ? 

9.  I  purchase  flour  at  $4.75  per  barrel ;  what  must  I  sell  it 
for  to  gain  12  per  cent.  ? 

10.  A  merchant  bought  a  ship  for  11475  dollars,  and  sold 
her  for  $13680  ;  what  did  he  gain  per  cent.  ? 

11.  The  population  of  the  State  of  New  York  in  1810,  was 
959949 ;  in  1820,  it  was  1372812  ;  what  was  the  gain  per, 
cent,  in  that  term  of  10  years  ? 

12.  In  1830,  it  was  1918604;  what  was  the  gain  per  cent, 
from  1820  to  1830  ? 

13.  In  1840,  it  was  2428921 ;  what  was  the  gain  per  cent, 
from  1830  to  1840  ? 

14.  The  population  of  Ohio  in  1810,  was  230760 ;  in  1820, 
it  was  581434 ;  what  was  the  gain  per  cent,  ? 

15.  The  population  of  Ohio  in  1830,  was  937903 ;  what 
was  the  gain  per  cent,  from  1820  to  1830  ? 

16.  In  1840,  it  was  1519467 ;  what  was  the  gain  per  cent, 
from  1830, to  1840  ? 

17.  Massachusetts  had,  in  1810,  472040  inhabitants;  in 
1820,  it  had  523287  ;  what  was  the  gain  per  cent,  in  10  years  ? 

18.  Massachusetts  had,  in  1830,  610408  inhabitants  ;  what 
was  her  gain  per  cent,  from  1820  to  1830  ? 

19.  In  1840,  Massachusetts  had  737699  inhabitants ;  what 
was  the  gain  per  cent,  from  1830  to  1840  ? 

20.  An  agent  sells  12000 '  dollars'  worth  of  cloth  for  a 
factory,  charging  2^  per  cent,  commission ;  what  will  be  his 
remuneration  ? 

21.  If  I  buy  for  a  merchant,  at  a  commission  of  4  per  cent., 
500  barrels  of  flour,  at  $4.40  per  barrel,  what  am  I  entitled 
to  for  my  commission  ? 

22.  What  is  3  per  cent,  on  $674.54? 

23.  What  is  2  per  cent,  on  $781.50  ? 

24.  What  is  the  value  of  five  100  dollar  shares  in  a  bank, 
at  4^  per  cent,  advance  ? 

25.  What  is  the  value  of  seven  100  dollar  shares,  at  6  per 
cent,  discount  ? 


FEB   CENTAGE,  167 

26.  "What  is  the  value  of  18  shares  bank  stock,  60  dollars 
a  share,  at  4  per  cent,  discount .'' 

27.  What  is  the  duty  on  a  quantity  of  broadcloth,  whose 
value  is  1735  dollars,  at  15  per  cent.  ? 

28.  What  is  the  duty  on  a  quantity  of  iron,  whose  value  is 
3456  dollars,  at  18  per  cent.? 

29.  What  is  the  commission  on  the  sale  of  1246  dollai's' 
worth  of  cloth,  at  3  per  cent.  ? 

30.  A  man  bought  a  lot  of  hay  for  13  dollars  a  ton ;  he 
sold  it  for  $14.25  a  ton  ;  what  did  he  gain  per  cent.  ? 

31.  Bought  tea  for  46  cents  a  pound ;  what  must  I  sell  it 
for  a  pound  to  gain  12  per  cent.  ? 

32.  What  is  the  worth  of  750  dollars,  bank  stock,  at  7^  per 
cent,  advance  ? 

33.  What  is  the  worth  of  8500  dollars,  bank  stock,  at  9  per 
cent,  discount  ?  ' 

34.  I  sell  flour  at  $5.32  per  barrel,  and  thereby  gain  12 
per  cent,  on  my  outlay,  what  did  the  flour  cost  ? 

Every  $1.00  laid  out  in  the  purchase  has  brought  me  a 
return  of  $1.12*;  the  number  of  dollars  I  paid  out  on  a  barrel 
must  therefore  equal  the  number  of  times  $1.12  will  go  in 
$5.32. 

35.  A  merchant  sells  a  ship  for  13680  dollars,  gaining 
thereby  14^^  per  cent,  on  what  she  coBt  him ;  what  did  the 
ship  cost  ? 

36.  300  dollars  is  2^  per  cent,  on  what  sum  ? 

37.  $15.63  is  2  per  cent,  on  what  sum  ?    . 

38.  Bought  12  barrels  of  flour,  each  containing  196  pounds, 
at  $5.42  per  barrel,  and  sold  it  at  26  cents  for  7  pounds ;  how 
much  gain  in  the  whole,  and  how  much  gain  per  cent.  ? 

39.  Bought  43  dozen  pairs  of  shoes,  at  $4.30  per  dozen, 
and  sold  them  at  62  cents  per  pair ;  what  gain  in  all?  what 
gain  per  cent.  ? 

40.  Bought  20  barrels  of  apples,  each  containing  2f  bushels, 
at  $2.10  per  barrel,  and  sold  them  at  $1.25  per  bushel ; 
What  gain  in  all  ?  what  gain  per  cent.  ? 

41.  Bought  375  barrels  of  flour,  at  $5.20  per  barrel,  and 
sold  200  barrels  at  $6.10,  the  remainder  at  $6.42  per  barrel ; 
What  gain  in  all  ?  what  gain  per  cent.  ?  ,. 

.  42.  Bought  34  acres  of  land,  at  41  4ifollars  per  acre ;  sold 
ifpit  $1700.00  ;  how  much  gain  in  all  ?  what  gain  per  qent.  ? 


168  ALLIGATION. 

SECTION    XXXIII. 

ALLIGATION.* 

The  operations  under  this  rule  show  the  method  of  finding 
the  value  of  a  mixture,  when  the  price  and  quantity  of  each 
of  its  ingredients  are  given  ;  also,  to  find  the  quantity  of  each 
ingredient,  when  its  price  is  given,  and  it  is  required  to  unite 
them  so  as  to  form  a  mixture  of  a  given  value. 

Case  1.  To  find  the  value  of  the  mixture,  when  the  quantity 
and  'price  of  each  of  the  ingredients  are  given. 

1.  Mix  15  bushels  of  oats,  at  40  cents  per  bushel;  12 
bushels  of  barley,  at  60  cents  ;  and  24  bushels  of  corn  at  83 
cents  ;  what  will  the  mixture  be  worth  per  bushel  ? 

It  is  evident  that  if  you  find  the  value  of  the  whole,  and 
divide  the  sum  by  the  number  of  bushels,  the  quotient  will  be 
the  value  per  bushel. 

2.  Mix  20  pounds  of  tea,  at  43  cents  per  pound ;  18  lbs.  at 
61  cents ;  and  11  lbs.  at  74  cents  per  pound;  what  will  the 
mixture  be  worth  ? 

3.  If  41  lbs.  of  coffee,  at  13  cents  per  lb.  be  mixed  with  45 
lbs.  at  9^  cents ;  and  27  lbs.  at  15  cents ;  what  will  the  mix- 
ture be  worth  per  pound  ? 

Gase  2.  To  find  the  quantity  of  each  ingredient,  when  its 
price  and  that  of  the  required  mixture  are  given. 

4.  K  I  mix  oats  worth  2s.  per  bushel,  with  rye  worth  5s., 
so  as  to  make  the  mixture  worth  3s.  per  bushel,  in  what  pro- 
portion must  I  mix  them  ? 

It  is  evident,  that  if  I  put  in  1  bushel  of  oats,  I  gain  1  shil- 
ling, f  Now  I  must  put  in  rye  enough  with  this  bushel  of  oats 
to  lose  (I i  shilling.  On  every  bushel  of  rye  put  in,  I  lose  2 
shillings  l! therefore,  in  order  to  lose  1  shilling,  I  must  put  in 
^  a  bushel.  ^  I  must  therefore  put  in  1  bushel  of  oats  to  ^  a 

*  The  word'-^j^gWifion  sigftifiBi^i^^j/???^  torjether ;  and  has  reference  to  a 
particular  way  O'^VWl'*^  ^^^"^ ^&rs  toget;iier,  by  means  of  which  operations 
of  this  kind  have'^^HM^lform'W^.  T^^name  is  retained  as  a  matter  of 
convenience;  but'^r^M^&bn;:]!!  it  J>^s^or  the  proj^rcss  of  tlie  pupil  that 
fce  should  pursue  a  sin^PJPl^glyticat  liit^if^^n  all  tlic  operations. 


ALLIGATION.  169 

bushel  of  rje.  It  is  evident  that,  if  I  double  the  quantity  thus 
found  of  each  ingredient,  the  value  of  the  mixture  will  be  the 
same  ;  or  I  may  take  any  equal  multiples  of  the  quantities,  as 
4  bushels  of  oats,  and  2  bushels  of  rye  ;  6  bushels  of  oats,  and  3 
bushels  of  rye  ;  20  bushels  of  oats,  and  10  bushels  of  rye,  &c. 

5.  If  I  mix  oats,  worth  2s.  per  bushel,  with  rye,  worth  6s., 
BO  as  to  make  the  mixture  worth  3s.  per  bushel ;  in  what  pro- 
portion must  they  be  mixed  ? 

6.  Mix  oats,  worth  3s.  per  bushel,  with  wheat,  worth  7s., 
so  as  to  make  the  mixture  worth  5s.  per  bushel ;  in  what  pro- 
portion must  they  be  mixed  ? 

7.  Mix  the  same  ingredients,  at  the  same  price,  so  as  to 
make  the  mixture  worth  6s.  per  bushel  j  in  what  proportion 
must  they  be  mixed  ? 

8.  In  what  proportion  must  oats,  worth  2s.,  and  wheat,  worth 
8s.,  be  mixed,  to  make  the  mixture  worth  4s.  per  bushel  ? 

9.  How  can  you  mix  corn,  worth  80  cents  per  bushel,  and 
rye,  worth  85  cents,  with  barley,  worth  46  cents,  so  as  to 
make  a  mixture  worth  60  cents  per  bushel  ? 

Here  you  have  three  ingredients.  First,  mix  barley  with 
one  of  the  dearer  ingredients,  so  as  to  make  a  mixture  of  the 
required  value ;  then  mix  barley  with  the  other  ingredient, 
and  see  how  much  you  have  taken  of  each. 

10.  Mix  3  sorts  of  tea,  at  25  cents,  33  cents,  and  40  cents 
per  lb.,  so  as  to  make  a  mixture  worth  30  cents  per  lb. 

11.  Mix  tea  at  20  cents,  with  tea  at  45  cents,  and  tea  at  54 
cents  per  lb.,  so  as  to  make  a  mixture  worth  38  cents  per  lb. 

12.  If  you  mix  sugar,  at  6  cents,  8  cents,  10  cents,  and  11 
cents  per  lb.,  in  what  quantities  may  they  be  taken,  so  as  to 
make  a  mixture  worth  9  cents  per  lb.  ? 

First,  take  two  of  the  ingredients,  one  cheaper  and  one 
dearer  than  the  mixture  ;  form  a  mixture  of  these  ;  then  take 
the  two  remaining  ingredients  in  the  same  way. 

13.  If  three  sorts  of  spirit,  worth  60  cents,  75  cents,  and  80 
cents  per  gallon,  are  mixed  with  water  costing  nothing,  what 
must  be  the  proportion  to  make^a  mixture  worth  70  cents  per 
gallon? 

It  is  immaterial  in  what  way  you  sele^the  pairs  of  ingre- 
dients, provided,  in  each  j)air,  one  of  the,ingredients  be  cheaper 
and  the  other  dearer  than  the  reqiured  mixture.    Thus  a  great 
15 


170  ALLIGATION. 

Tariety  of  answers  may  be  obtained  whenever  there  is  more 
than  one  pair  of  ingredients.  In  all  cases,  however,  the  cor- 
rectness of  the  operation  may  be  proved  in  the  following  way : 
Find  the  total  value  of  all  the  ingredients  ;  if  this  is  equal  to 
the  value  of  the  whole  mixture  at  the  required  price,  the  work 
is  right. 

14.  Mix  5  sorts  of  grain,  at  25  cents,  30  cents,  33  cents, 
45  cents,  and  50  cents,  so  as  to  make  a  mixture  worth  40 
cents  per  bushel. 

Case  3.    When  the  quantity  of  one  ingredient  is  given. 

15.  Mix  brandy,  at  74  cents  per  gallon,  with  24  gallons  of 
brandy,  at  1  dollar  per  gallon,  so  that  the  mixture  may  be 
worth  80  cents  per  gallon. 

Here  you  observe  that  the  quantity  of  one  of  the  ingredients 
is  given.  We  will  first  make  a  mixture  of  the  two,  without 
regard  to  this  circumstance.  If  I  put  in  1  gallon  at  1  dollar, 
I  lose  20  cents.  For  every  gallon  at  74  cents,  which  is  put 
in,  I  gain  6  cents.  In'  order  to  gain  20  cents,  I  must,  there- 
fore, put  in  3^  gallons.  The  quantities  stand,  then,  1  gallon 
at  1  dollar,  3^  gallons  ,at  80  cents.  But  I  wish  to  put  in  24 
gallons  at  1  dollar.  To  balance  this,  I  must,  therefore,  put  in 
24  times  3^  gallons,  at  74  cents  ;  that  is  80  gallons. 

16.  Mix  sugar,  at  8  cents,  11  cents,  and  12  cents,  with  100 
lbs.  of  sugar,  at  7  cents,  so  as  to  make  the  mixture  worth  10 
cents  per  lb. 

Case  4.  When  the  quantity  of  the  required  mixture  is  given. 

17.  Mix  oats,  at  40  cents,  with  corn,  at  60  cents,  so  as  to 
form  a  mixture  of  100  bushels,  worth  48  cents  per  bushel. 

If  I  put  in  1  bushel,  at  40  cents,  I  gain  8  cents ;  if  I  put  in 
1  bushel,  at  60  cents,  I  lose  12  cents.  To  lose  8  cents,  there- 
fore, I  must  put  in  only  §  of  a  bushel.  The  qiiantities  are, 
then,  1  bushel  at  40  cents,  f  of  a  bushel  at  60  cents  ;  making, 
when  added.  If  bushels.  But  100  bushels  is  the  quantity 
required.  100-^-^=60.  Each  ingredient,  therefore,  must  be 
multiplied  by  60  ;  60X1=60  ;  60  Xf=40.-  The  quantities, 
then,  are  60  bushels  at  40  cents,  and  40  bushels  at  60  cents. 


EQUATION   OP  PAYMENTS.  17-1 

SECTION    XXXIV. 

EQUATION    OF    PAYMENTS. 

If  A  owes  B  several  sums  of  money  to  be  paid  at  differ- 
ent times,  he  may  desire  to  pay  the  -^hole  at  once,  and  con- 
sequently to  know  at  what  time  the  whole  becomes  due. 
This  time  is  found  by  making  an  equation  of  the  paymerUs, 
multiplied  by  the  time,  as  follows. 

1.  A  owes  B  200  dollars;  100  due  Jan.  1,  1844;  100  due 
Jan.  1,  1846 ;  he  wishes  to  pay  it  all  at  once.  At  what  time 
should  he  pay  it  ? 

Now,  on  Jan.  1,  1844,  A  is  entitled  to  the  use  of  100  dol- 
lars for  2  years  longer;  100X2=200 ;  equal  to  the  use  of  1 
dollar  for  200  years.  If  he  is  to  pay  the  whole  together,  he 
must  keep  the  200  dollars  long  enough  to  balance  that  claim ; 
200)200(1  year,  —  the  answer.  The  whole  should  be  paid 
one  year  from  Jan.  1,  1844. 

2.  A  owes  B  100  dollars,  due  in  6  months ;  200  dollars 
due  in  12  months  ;  in  how  many  months  should  the  whole  be 
paid  together  ? 

100  X   6=  600 
200X12=2400 

300:   300)3000 

10  months ;  the  answer. 

The  above  is  the  method  usually  employed,  and  is  suffi- 
ciently exact  for  the  necessities  of  business ;  but  it  gives  a 
result  a  little  in  favor  .of  the  debtor;  that  is,  it  makes  the 
equated  time  a  little  later  than  it  should  be.  To  find  the 
exact  equated  time  is  a  problem  far  too  difficult  to  be  used  in 
ordinary  business. 

Rule.  Multiply  each  payment  by  the  length  of  time  be- 
fore it  becomes  due.  Divide  the  sum  of  the  products  by  the 
sum  of  all  the  payments ;  the  quotient  will  express  the  length 
of  time  in  which  the  whole  is  due. 

3.  A  owes  B  several  sums,  due  at  different  times,  as  fol- 
lows ;  $600  in  2  months,  $150  in  3  months,  $75  in  6  months; 
what  is  the  equated  time  for  the  whole  ?  • 

4.  A  man  owes  $1000 ;  of  which,  200  are  to  be  paid  in  3 


172  SQUARE   MEASURE. 

montlis,  400  In   6  montlis,  and  the  remainder  in  8  months ; 
what  is  the  equated  time  for  the  payment  of  the  whole  ? 

5.  If  I  owe  $1200,  one  half  to  be  paid  in  3  months,  one 
third  in  6  months,  and  the  remainder  in  9  months ;  in  what 
time  should  the  whole  be  paid  ? 

6.  A  owes  B  $640 ;  150  due  in  30  days,  200  due  in  60 
days,  and  the  remainder  in  90  days  ;  what  is  the  equated 
time  for  the  whole  ? 

7.  A  merchant  buys  goods  to  the  amount  of  $1800 ;  one 
third  to  be  paid  in  30  days,  one  third  in  45  days,  and  the  re- 
mainder in  90  days  ;  what  is  the  equated  time  for  the  whole  ?^ 

8.  If  I  owe  $1000,  half  to  be  paid  in  60  days,  and  half 
in  120  days,  and  if  I  pay  $100  down,  what  will  be  the 
equated  time  for  the  remainder  ? 


SECTION    XXXV. 

SQUAEE    MEASURE. 
[See  Section  XV.  Part  I.] 

1.  There  is  a  field  in  the  form  of  a  square,  15  rods  on  a 
side ;  how  many  square  rods  does  it  contain  ? 

2.  If  the  square  be  15^  rods  on  a  side,  how  many  square 
rods  will  it  contain  ? 

3.  How  many  square  rods  are  there  in  a  square  field  meas- 
uring 17  rods  on  a  side  ? 

4.  If  the  field  measure   17^  rods   on   a   side,  how  many 
square  rods  will  it  contain  ? 

5.  What  is  the  contents  of  a  square  field  measuring  21^ 
rods  on  a  side  ? 

6.  What  is  the  area  of  a  rectangular  field,  its  length  being 
64  rods,  and  its  breadth,  12|  rods? 

7.  There  is  a  rectangular  field,  its  dimensions  being  24^ 
rods,  and  76^  rods  ;  what  is  the  area  ? 

8.  How  many  acres-  are  there  in  a  rectangular  field,  its  di- 
mensions being  94  rods  and  76^  rods? 

9.  There  is  a  rectangular  field   containing  7  acres  ;   its 
length  is  35  rods ;  what  is  its  breadth  ? 


SQUARE   MEASURE.  178 

10.  There  is  a  rectangular  farm ;  its  length  being  132  rods  j 
its  breadth,  86 J;  how  many  acres  does  it  contain? 

11.  There  is  a  rectangular  lot  of  land  containing  325 
acres ;  it  measures  on  one  side  176  rods ;  what  will  it  meas- 
ure on  the  other  ?  . 

12.  There  is  a  board  containing  12  square  fpet;  it  is  13 
inches  wide ;  how  long  is  it  ? 

13.  A  table  contains  15  square  feet ;  it  is  4  feet  long ;  how 
wide  is  it? 

14.  A  certain  room  contains  30  square  yards;  it  is  16  feet 
wide ;  how  long  is  it  ? 

15.  A  piece  of  cloth  is  If  yards  wide ;  how  much  in  length 
will  it  require  to  make  8  square  yards  ? 

16.  There  is  a  room  15  feet  by  18;  how  many  yards  of 
carpeting,  ^  of  a  yard  wide,  will  it  require  to  cover  it  ? 

17.  How  many  feet  of  boards  will  it  require  to  cover  the 
sides  and  ends  of  a  barn,  as  high  as  to  the  eaves,  —  its 
length  is  42  feet,  width  34,  and  height  18,  —  allowing  one 
fifth  of  the  boards  to  be  wasted  in  cutting  ? 

18.  What  will  the  above-named  amount  of  boards  cost  at 
$11.50  a  thousand  feet? 

19.  A  road  3^  rods  wide,  passes  through  a  man's  land  1 
mile ;  how  much  of  his  land  does  it  take  ? 

20.  To  what  damages  will  he  be  entitled,  allowing  him  28 
dollars  an  acre  ? 

21.  There  is  a  right  angled  triangle ;  its  base  is  64  rods, 
and  perpendicular  20  rods ;  how  many  acres  does  it  contain  ? 
(See  Sec.  XVII.  Pt.  I.) 

22.  There  is  a  right  angled  triangle ;  its  base  is  84  rods, 
and  perpendicular  26  rods  ;  how  many  acres  does  it  contain? 

23.  There  is  a  right  angled  triangle ;  its  base  is  49  rods, 
perpendicular  34  rods ;  how  many  acres  does  it  contain  ? 

24.  There  is  a  right  angled  triangle  ;  its  area  is  640  rods ; 
the  base  is  64  rods  ;  what  is  the  perpendicular  ? 

25.  A  right  angled  triangle  has  an  area  of  1092  rods ;  its 
base  is  60  rods ;  what  is  the  perpendicular  ? 


15* 


174 


DUODECIMALS. 


SECTION    XXXVI 


DUODECIMALS. 


In  measuring  wood  and  lumber,  the  dimensions  are  taken 
in  feet  and  inches.  As  one  inch  is  -ji^-  of  a  foot,  the  multipli- 
cation of  feet  and  inches  by  feet  and  inches,  is  the  same  as 
multiplying  integers  and  twelfths  by  integers  and  twelfths. 
Take  the  following  example  : 

Operation. 

2     A 


1.  What  is  the  contents  of  a 
board  3  feet  7  inches  long,  and 
2  feet  4  inches  wide  ? 


Ans.    6     ft     T^. 


This  answer  may  be  reduced  to  more  simple  terms.  -^^^ 
=i^+tI¥  ;  adding  T^j-f  l=f  l»  and  this  again  =  2  feet-|- 
■^ ;  adding  the  2  feet  to  the  6  feet,  the  answer  stands  8  feet 

As  the  fractions  decrease  in  value  at  a  twelve  fold  rate, 
whenever  the  numerator  exceeds  12,  the  excess  may  be  set 
down,  and  the  one  or  more  carried  to  the  next  higher  fraction. 


2.  Multiply  5  feet,  2  inches, 
by  11  feet,  9  inches. 


5 
11 


T^ 

rW 


3 

56 


it 
if 


tU 


Ans.     60     ^    tI^. 


To  render  the  operation  more  simple,  call  the  12ths  or 
inches,  primes,  (marked  ')  and  the  144ths  or  fractions  of  the 
second  order,  seconds,  (marked  " ;)  then  begin  with  the  lowest 
order  and  multiply,  setting  each  product  in  its  own  place, 
with  the  mark  appropriate  to  express  its  value. 


3.  Multiply  13  ft.  5  in.,  by  2  ft. 
11  in. 


ft. 

/ 

13 

5 

2 

11 

12 

3' 

7" 

26 

10 

Ans.    39      1'     7" 


EXTRACTION  OF  THE  SQUARE  ROOT.       175 

4.  Multiply  3  ft.  9  in.  by  7  ft.  4  in. 

5.  Multiply  9  ft.  8  in.  by  4  ft.  9  in. 

6.  Multiply  15  ft.  2  in.  by  9  ft.  1  in. 

7.  Multiply  8  ft.  6  in.  by  2  ft.  4  in. 

8.  What  is  the  contents  of  a  board,  14  ft.  5  in.  long,  and 

1  ft.  1  in.  wide  ? 

9.  How  many  feet  in  a  load  of  wood,  8  ft.  6  in.  long,  4  ft. 

2  in.  wide,  and  3  ft.  7  in.  high  ? 

Multiply  two  of  the  dimensions  together,  and  that  product 
by  the  third  dimension. 

10.  How  much  wood  in  a  load  11  ft.  3  in.  long,  4  ft.  4  in. 
wide,  3  ft.  11  in.  high? 

Divide  the  cubic  feet  by  128  for  cords,  and  the  remainder 
by  16  for  cord  feet,  or  eighths  of  a  cord. 

11.  How  much  wood  in  a  pile  38  ft.  6  in.  long,  4  ft.  2  in. 
wide,  and  4  ft.  high  ? 

12.  How  much  wood  in  a  load  9  ft.  4  in.  long,  4  ft.  3  in. 
'wide,  3  ft.  8  in.  high  ? 

13.  How  much  wood  in  a  load  7  ft.  8  in.  long,  4  ft.  2  in. 
wide,  3  ft.  4  in.  high  ? 

14.  How  much  wood  in  a  load  8  ft  2  in.  long,  4  ft.  wide, 
4  ft.  3  in.  high? 

15.  How  many  cords  of  wood  will  a  shed  contain,  whose 
dimensions  inside  are  22  ft.  6  in.  long,  10  ft.  6  in  wide,  7  ft. 
8  in.  high  ? 

16.  Three  men  own  equal  shares  in  a  lot  of  wood  lying  in 
two  piles ;  one  pile  is  13  ft.  4  in  long,  4  ft.  3  in.  wide,  4  ft.  4 
in.  high ;  the  other  pile  is  17  ft.  long,  4  ft.  wide,  3  ft.  10  in. 
high ;  how  much  wood  is  each  man  share  ? 

See  note  on  page  84. 


SECTION    XXXVII. 

EXTRACTION  OF  THE  SQUARE  ROOT, 
CSee  Section  XVI.  Part  I.] 

This  operation  will  be  best  understood,  by  talking  first  the 
simplest  case,  where  the  number  is  an  exact  square,  and  the 
root  containing  only  two  figures. 


176 


EXTRACTION  OF  THE  SQUARE  ROOT. 


What  is  the  square 
root  of  196? 


Operation. 


196 
100 


20)' 


96 

80 
16 
96 
00 


10  1st  part  of  the  root. 
4  2d  part  of  the  root. 

Ans. 


14 


Place  a  period  over  the  unit  figure ;  another  OA^er  that  of 
hundreds.  This  will  show  how  many  figures  there  will  be  in 
the  root ;  for  the  square  of  a  number  has  always  either  twice 
as  many  figures  as  the  number,  or  one  less  than  twice  ai 
many.  Find  the  greatest  square  of  tens  in  the  first  period, 
(in  the  given  example,  100,)  and  set  its  root  (10)  in  the 
quotient.  This  will  be  the  first  part  of  the  root.  Square 
the  root ;  subtract  the  square  from  the  first  period,  and  bring 
down  the  figures  of  the  next  period  for  a  dividend.  To  the 
left  hand,  place  double  the  part  of  the  root  already  found  for 
a  trial  divisor.  Find  by  trial,  what  the  next  figure  of  the 
root  must  be,  and  set  it  down  under  the  first  part  of  the  root. 
This  is  the  second  part,  or  unit  figure  of  the  root.  [In  try- 
ing for  this  figure,  remember,  that  it  must  be  so  small  that 
when  the  divisor  shall  be  multiplied  by  it,  and  the  square  of 
itself  shall  be  added  to  the  product,  the  sum  shall  not  ex- 
ceed the  dividend.]  Multiply  the  divisor  by  the  new  figure 
of  the  root;  to  this  add  the  square  of  the  same  figure,  and 
subtract  the  sum  from  the  dividend.  If  the  number  is  an 
exact  square  of  two  periods,  as  in  the  above  example,  there 
M'ill  be  no  remainder;  and  the  two  parts  of  the  root  thus 
found,  when  added  together,  will  give  the  whole  root. 

2.  What  is  the  square  root  of  225  ?     Of  324  ? 

3.  What  is  the  square  root  of  289  ? 

4.  What  is  the  square  root  of  361  ? 

5.  Wliat  is  the  square  root  of  625  ? 

6.  What  is  the  square  root  of  784 ? 

7.  Wliat  is  the  square  root  of  841  ? 

8.  Wliat  is  the  square  root  of  961  ? 
If  there  are  more  than  two  periods,  first  consider  only  the 

two  left  hand  periods,  and  find  their  root  as  above  directed ; 
then  consider  the  part  of  the  root  expressed  by  these  two 


Of  529? 
Of  729  ? 
Of  1024? 
Of  1296? 
Of  1849? 
Of  2601 ? 


EXTRACTION  OF  THE  SQUARE  ROOT.       177 

figures  as  the  first  part  with  reference  to  the  next  figure,  (to 
indicate  this,  you  must  annex  a  cipher,)  and  work  for  the 
next ;  and  so  on. 

9.  What  is  the  square  root  of  15625  ? 

10.  What  is  the  square  root  of  60516?  Of  104976? 
Of  2134M  ? 

Square  Root  of  a  Decimal. 
If  there  are  decimals  in  the  number,  point  oflp  each  way 
from  the  place  of  units ;   adding  a  cipher,  if  necessary,  to 
make  the  right  hand  period  complete. 

11.  What  is  the  square  root  of  2.56  ?     Of  12.25  ? 

12.  What  is  the  square  root  of  2.25  ?     Of  20.25  ? 

13.  What  is  the  square  root  of  156.25  ?     Of  132.25  ? 

14.  What  is  the  square  root  of  13.6^?     Of  21.16? 

15.  What  is  the  square  root  of  88.36?     Of  53.29  ? 

16.  What  is  the  square  root  of  1.69  ?     Of  1.44? 

17.  What  is  the  square  root  of  .81  ?     Of  .64? 

18.  What  is  the  square  root  of  .01  ?     Of  6.25  ? 

Square  Root  of  a  Vulgar  Fraction. 
To  obtain  the  square  root  of  a  vulgar  fraction,  find  the 
square  root  of  the  numerator,  and  of  the  denominator,  and 
write  the  former  over  the  latter. 

19.  What  is  the  square  root  of  f  ?     Of  if  ? 

20.  What  is  the  square  root  of  ^  ?     Of  ^  ? 

21.  What  is  the  square  root  of  i|f  ?     Of  ||  ? 

22.  What  is  the  square  root  of  |||  ?     Of  |||  ? 

The  correctness  of  the  answer  may  always  be  tested  by 
multiplying  the  answer  found,  by  itself;  if  correct,  it  will  re- 
produce the  original  square. 

23.  What  is  the  square  root  of  ^  ?     Of  ^  ? 

^  24.  What  is  the  square  root  of  if?     Of  t^V? 

Another  Method  of  finding  the  Root  of  a  Fraction.  Re- 
duce the  fraction  to  a  decimal,  and  proceed  as  already  directed 
in  the  case  of  decimals. 

25.  What  is  the  square  root  of  ^  ?  The  square  root  of  1 
is  1,  the  square  root  of  4  is  2  ;  ans.  ^ ;  or  reduce  |^  to  a  deci- 
mal, =  .25  ;  square  root,  .5,  answer. 

If  the  number  is  not  a  complete  square,  annex  periods  of 
ciphers,  as  decimals,  and  carry  the  operation  as  far  as  desired. 


178       EXTRA.CTION  OF  THE  SQUARE  ROdT. 

26.  What  ia^flie  square  root  of  70  ?     Of  80  ? 

27.  What  is  the  square  root  of  90  ?     Of  45  ? 

28.  What  is  the  square  root  of  60?     Of  84? 

29.  What  is  the  square  root  of  200  ?     Of  120  ? 

30.  There  is  a  field  in  the  form  of  a  square,  containing  1 
acre  ;  how  many  rods  does  it  measure  on  a  side  ? 

31.  There  is  a  right  angled  triangle  ;  its  hypotenuse  meas- 
uring 60  rods.  What  is  the  sum  of  the  squares  of  the  two 
legs  ?     (See  Sec.  XVII.  Part  I.) 

32.  There  is  a  right  angled  triangfe ;  the  squares  of  its 
legs  added  together  are  81  rods  ;  what  is  the  length  of  the 
hypotenuse  ? 

33.  There  is  a  right  angled*  triangle ;  its  legs  measure,  — 
one  25,  the  other  30  rods  ;  how  long  is  the  hypotenuse  ? 

34.  Two  men  start  from  the  same  place;  one  travels  8 
miles  east ;  the  other,  15  miles  north  ;  how  far  are  they  then 
apart? 

35.  A  ladder  40  feet  long  stands  against  a  house,  the  foot 
resting  on  the  ground,  on  a  level  with  the  foundation  of  the 
house,  and  20  feet  distant  from  it ;  how  far  up  will  it  reach  ? 

36.  The  floor  of  a  room  measures  16  feet  in  length,  and  14 
feet  in  width ;  how  long  a  line  will  reach  diagonally  from 
corner  to  corner  ? 

37.  The  two  parts  of  a  carpenter's  square,  one  12,  the  other 
24  inches  long,  may  be  regarded  as  the  legs  of  a  right  angled 
triangle ;  how  long  would  be  the  hypotenuse  connecting  their 
extremities  ? 

38.  There  is  a  room  16  feet  long,  14  feet  wide,  and  10  feet 
high ;  how  long  must  a  straight  line  be,  reaching  from  a 
corner  of  the  room,  at  the  bottom,  to  the  diagonal  corner,  at 
the  top  ? 

39.  There  is  a  room,  the  length,  breadth,  and  height  of  which 
are  each  10  feet ;  how  far  is  it  from  a  corner  of  the  room  at 
the  bottom,  to  the  diagonal  corner  at  the  top  ? 

40.  There  is  a  room,  the  length,  breadth,  and  height  of  which 
are  equal ;  the  distance  from  a  corner,  at  the  bottom,  to  the 
diagonal  corner  at  the  top,  is  18  feet ;  what  is  the  size  of  the 
room? 

41.  I  have  a  cubic  block,  measuring  4  inches  each  way ; 
how  far  apart  are  its  diagonal  corners  ? 

42.  How  large  a  cube  can  be  cut  from  a  sphere  which  is  1 
foot  in  diameter  ? 


\ 


EXTRACTION  OF  THE  CUBE  ROOT. 


179 


SECTION    XXXVIII 


EXTRACTION  OF  THE  CUBE  EOOT. 
[See  Section  XIX.  Part  I.] 

We  will  first  consider  those  numbers  tlie  cube  root  of  which 
is  expressed  by  a  single  figure.  Every  exact  cube  of  not  more 
than  three  figures,  will  have  for  its  root  some  number  less 
than  10,  and,  consequently,  it  will  be  expressed  by  a  single 
figure.     This  root  can  be  found  by  successive  trials. 

Examples. 

1.  What  is  the  cube  root  of  125  ?  .    , 

2.  What  is  the  cube  root  of  216  ? 

3.  What  is  the  cube  root  of  512  ? 

4.  What  is  the  cube  root  of  729  ? 

We  will  next  take  perfect  cubes,  the  root  of  wluch  consists 

of  two  figures.  operation. 


300 
30 

4096 
1000 

10,  1st  part  of  the  root. 
6,  2d  part  of  tlie  root. 

5.  What  is  the  cube 

330 

3096 

16,  Answer. 

root  of  4096? 

1800 

1(?80 

216 

3096 

0000 

Rule.  Place  a  .period  over  the  unit  figure,  and  another 
over  that  of  thousands.  Find  the  greatest  cube  in  the  first 
period,  whose  root  is  expressible  in  tens.  Set  down  this  root 
as  a  quotient  in  division ;  find  the  cube  of  the  root,  and  sub- 
tract it  from  the  first  period,  and  bring  down  the  second  period 
as  a  dividend.  At  the  left  hand  of  this  set  down  three  times 
the  square  of  the  root,  and  under  this  three  times  the  root ; 
add  these  together,  for  a  trial*  divisor.  Find,  by  trial,  what  the 
next  figure  of  the  root  will  be,  and  set  it  under  the  first  part 
already  found.  Multiply,  by  this  figure,  three  times  the  square 
of  the  first  part  of  the  root,  setting  the  product  under  the 
dividend.  Multiply,  by  the  square  of  this  figure,  three  times 
the  first  part  of  the  root,  setting  the  product  underneath  the 


180         EXTRACTION  OF  THE  CUBE  KOOT. 

other ;  under  these  set  the  cube  of  the  root  figure  last  found. 
Add  these  three  numbers  together,  and  subtract  their  sum 
from  the  dividend.  If  the  work  be  correct,  there  will  be  no 
remainder.  Add  together  the  two  parts  of  the  root  for  the 
answer. 

6.  What  is  the  cube  root  of  2744?     Of  205379  ? 

7.  What  is  the  cube  root  of  3375  ?     Of  -5832  ? 

8.  What  is  the  cube  root  of  4913  ?     Of  10648  ? 

9.  What  is  the  cube  root  of  9261  ?     Of  15625  ? 

10.  What  is  the  cube  root  of  13824?     Of  19683  ? 

11.  What  is  the  cube  root  of  46656  ?     Of  39304  ? 

We  will  next  consider  the  case  where  there  are  more  than 
two  figures  in  the  root.  The  number  of  figures  in  the  root 
can  always  be  determined  by  the  number  of  periods  placed 
over  the  sum,  beginning  with  units,  and  placing  a  period  over 
every  third  place.  If  there  are  more  than  three  periods  in 
the  cube,  regard,  first,  only  the  two  left  hand  periods,  obtain- 
ing the  first  and  second  figures  of  the  root,  just  as  if  they 
constituted  the  whole  root.  Then,  after  bringing  down  the 
figures  of  another  period,  add  the  two  parts  of  the  root,  and 
consider  their  sum  as  the  first  part  of  the  root,  and  proceed  to 
find  the  next  part.  To  indicate  this,  you  must  annex  a  cipher 
to  the  figures  of  the  root  already  found. 

12.  What  is  the  cube  root  of  1953125  ? 

13.  What  is  the  cube  root  of  2406104  ? 

14.  What  is  the  cube  root  of  3796416  ? 

If  there  are  decimals  in  the  given  sum,  point  ofi*  both  ways 
from  the  units'  place,  adding  ciphers,  if  necessary,  to  the  deci- 
mal, in  order  to  make  the  period  complete. 

1.5.  What  is  the  cube  root  of  15.625  ? 

16.  What  is  the  cube  root  of  35.937  ? 

'  If  the  number  given  is  not  a  perfect  cube,  add  periods  of 
ciphers,  and  carry  out  the  root  in  decimals  as  far  as  may  be 
desired. 

17.  What  is  the  cube  root  of  10  ? 

18.  What  is  the  cube  root  of  20  ? 

19.  What  is  the  cube  root  of  50  ? 

20.  What  is  the  cube  root  of  100  ? 

21.  A  bushel,  even  measure,  contains  2152  solid  inches ; 
what  would  be  the  inside  measure  of  a  cubic  box  containing 
12  bushels  ? 


PROrORTION.  181 

22.  A  gallon,  wine  measure,  contains  231  cubic  inches ; 
what  must  be  the  inside  measure  of  a  cubic  cistern  containing 
10  barrels  ? 

23.  What  would  be  the  measure  of  a  cubic  pile  of  wood, 
containing  one  cord  ? 


SECTION    XXXIX. 

'  PEOPORTION. 
[See  Section  XX.  Part  I.] 

Several  changes  that  may  be  made  in  the  terms  of  a  Pro- 
portion, are  exhibited  in  page  105.  In  continuing  the  subject, 
we  will  first  state  some  further  changes  that  may  be  made  in 
the  terms  without  destroying  the  proportion. 

1.  Multiply  all  the  terms  by  the  same  number. 

2.  Divide  all  the  terms  by  the  same  number. 

3.  Add  the  terms  of  the  first  ratio  for  the  first  antecedent, 
and~the  terms  of  the  second  ratio  for  the  second  antecedent. 

4.  Add  the  terms  of  the  first  ratio  for  the  first  consequent, 
and  the  terms  of  the  second  ratio  for  the  second  consequent. 

5.  Instead  of  the  sum  of  the  terms  in  the  third  case  above, 
take  the  difference  of  the  terms. 

6.  Instead  of  the  sum  of  the  terms  in  the  fourth  case'  above, 
take  the  difference  of  the  terms. 

7.  Raise  each  term  to  the  same  power,  as  second  or  third 
power. 

8.  Extract  of  each  term  the  same  root. 

The  result,  after  each  of  these  operations,  will  still  be  a 
proportion,  and  may  be  proved  to  be  so,  by  multiplying  the 
extremes  together,  and  finding  the  product,  equal  to  that  of 
the  means. 

Take  the  proportion,  4  :  16  : :  9  :  36,  and  perform  on  it  the 
first  change,  using  any  number  you  please  for  a  multiplier, 
and  then  prove  the  proportion. 

Perform  on  the  same  proportion  the  second  change. 

Perform  the  third  change. 

Perform  the  fourth  change. 

Perform  the  fifth  change. 
16 


182  ^  PROrORTION.  ' 

Perform  tlie  sixth  change. 

Perform  the  seventh  change,  raising  to  the  second  power. 

Perform  the  eighth  change,  extracting  the  square  root. 

Finally,  you  may,  iu  any  case,  invert  the  whole  proportion; 
or,  invert  the  terms  of  each  ratio  ;  or  invert  the  means,  or  the 
extremes. 

Practical   Questions. 

1.  If  7  lbs.  of  flour  cost  31  cents,  what  will  196  lbs.  cost  ? 
As  the  smaller  quantity  is  to  the  larger  quantity,  so  is  the 

price  of  the  smaller  quantity  to  the^rice  of  the  larger. 

2.  If  3  cwt.  of  hay  cost  2  dollars,  what  will  35  cwt.  cost  ? 

3.  If  4  qts.  of  molasses  cost  38  cents,  what  will  10  qts.  cost? 

4.  If  a  horse  travels  19  miles  in  3  hours,  how  far  will  he 
travel  in  11  hours  ? 

5.  ir^he  freight  of  7  cwt.  cost  2  dollars,  what  will  the 
freight  of  20  cwt.  cost  ? 

6.  If  11  dollars  buy  3  cords  of  wood,  how  many  cords  will 
50  dolla;:s  buy  ? 

7.  If  7  bushels  of  oats  last  a  horse  2  months,  how  long  will 
23  bushels  last  him,  at  the  same  rate  ? 

8.  A  man  bought  a  horse  for  84  dollars,  and  sold  him  for 
$93  ;  what  did  he  gain  per  cent.  ? 

As  the  whole  outlay  is  to  1  dollar,  so  is  the  whole"  gain  to 
the  gain  on  a  dollar. 

9.  A  merchant  buys  flour  at  $4.35  a  barrel,  and  sells  it  for 
$4.63  ;  what  is  his  gain  per  cent.? 

10.  A  and  B  form  a  partnership  in  trade ;  A  puts  in  $500, 
and  B  $300,  for  the  same  time ;  they  gain  $180  ;  what  ought 
each  to  share  ? 

As, the  whole  stock  is  to  each  one's  share,  so  is  the  whole 
gain  to  each  one's  gain. 

.  11.  C  and  D  trade  in  company;  C  puts  in  750  dollars,  and 
D  $450,  for  the  same  time ;  they  gain  240  dollars ;  how  much 
gain  ought  each  to  receive  ? 

12.  Two  men  buy  a  lot  of  wood  in  company  for  340  dollars ; 
one  takes  away  42  cords,  the  other  the  remainder,  which  was 
34  cords  ;  what  ought  each  to  pay  ? 

13.  T'wo  men  hire  a  sheep-pasture  in  company  for  20, 
dollars ;  one  keeps  30  sheep  in  it  14  weeks ;  the  other  24 
sheep,  16  weeks  ;  what  ought  each  to  pay  ? 

Find  how  many  weeks'  pasturing  for  a  single  sheep  each 
one  had.  * 


PROPORTION.      SIMILAR    SURFACES.  183 

14.  Two  men  purchase  a  lot  of  standing  grass  for  S3 6.50  ; 
one  takes  3^  tons,  the  other  If  tons ;  what  ought  each  to  pay  ? 

Reduce  the  quantity  of  hay  to  fourths  of  a  ton,  and  then 
state  the  proportion. 

15.  There  is  a  circular  piece  of  ground,  whose  diameter  is 
14  rods ;  what  will  be  the  diameter  of  a  circle  containing 
twice  as  much? 

16.  There  is  a  circular  piece  of  ground  containing  2.5 
acres ;  what  will  be  the  area  of  a  circle,  the  diameter  of 
which  is  3  times  as  great  ? 

17.  There  are  two  similar  triangular  fields* ;  the  smaller 
contains  3  acres,  the  larger  4 ;  the  base  of  the  smaller  is  44 
rods  ;  how  long  is  the  base  of  the  larger  ? 

18.  There  are  two  similar  rectangular  fields ;  the  smaller  is 
34  rods  wide,  and  60  rods  long;  the  other  has  twice  as  great 
an  area ;  what  are  its  dimensions  ? 

19.  There  is  a  grindstone  4  feet  in  diameter;  what  will  be 
its  diameter  after  half  of  it  is  ground  ofi"? 

20.  There  are  two  similar  triangular  pieces  of  land ;  the 
base  of  one  measures  44  rods ;  the  other  piece  has  an  area  7 
times  as  large  as  the  first ;  what  is  the  length  of  its  base  ? 

21.  There  are  two  cisterns  of  the  same  shape ;  one  is  5 
'  feet  deep ;  the  other  has  a  capacity  three  times  as  great ;  how 

deep  is  it  ? 

22.  If  a  ball  5  inches  in  diameter,  weighs  14  lbs.,  what 
will  be  the  weight  of  one  of  the  same  material  6  inches  in 
diameter  ? 

23.  What,  on  the  same  supposition,  will  be  the  weight  of  a 
ball  of  7  inches  diameter  ? 

24.  There  are  two  marble  statues  of  the  same  form,  but 
differing  in  size  ;  one  is  5  feet  high,  and  weighs  740  lbs. ;  the 
other  is  7  feet  high  ;  what  will  it  weigh  ? 

25.  If  a  tree  2^  feet  in  diameter  at  the  ground,  contains  3 
cords  of  wood ;  how  much  will  there  be  in  a  tree  of  the  same 
-shape,  S^  feet  in  diameter? 

26.  There  are  two  similar  stacks  of  hay;  the  smaller  is 
11^  feet  high,  and  contains  4^  tons  of  hay;  the  larger  is  14 
feet- high  ;  how  much  hay  does  it  contain,  supposing  both  to 
be  of  the  same  solidity  ? 

27.  If  an  iron  field  piece  5^  feet  long,  weighs  1140  lbs., 


184  PROPORTION. 

how  many  lbs.  will  an  iron  cannon  of  the  same  shape  weigh, 
that  is  lOf  feet  long? 

28.  There  are  two  anchors  of  similar  form ;  the  smaller 
weighs  1100  lbs.,  the  larger  is  2}  times  as  long;  what  is  its 
weight  ? 

When  a  cause  and  an  effect  are  connected  together,  the 
increase  of  the  one  is  always  connected  with  an  increase  of 
the  other.  If  6  horses  eat  20  bushels  of  oats,  we  may  regard 
the  horses  as  the  cause,  and  the  consumption  of  the  oats  the 
effect ;  or,  if  we  please,  we  may  regard  the  oats  as  the  cause, 
and  the  support  of  the  horses  as  the  effect.  But  in  either 
case,  an  increase  of  one  would  require  an  increase  of  the 
other.  When  numbers  are  connected  in  this  way,  in  a  pro- 
portion, having  the  relation  of  cause  and  effect  to  each  other, 
the  proportion  is  said  to  be  Direct. 

But  it  often  happens,  that  quantities  are  connected  togeth- 
er, not  as  cause  and  effect,  but  as  limitations  of  each  other ; 
where  an  increase  of  one  quantity  requires  a  diminution  of 
the  other. 

Thus,  if  the  provisions  of  a  ship's  company  are  sufficient  to 
last  17  weeks  at  the  rate  of  13  oz.  of  bread  per  day  for  each 
man,  it  is  evident  that  these  quantities,  17  and  13,  are  not 
cause  and  effect,  but  limitations  of  each  other.  If  one  is  in- 
creased, the  other  must  be  diminished.  So,  if,  with  a  speed 
of  11  miles  per  hour,  a  journey  be  performed  in  31  hours,  it 
is  evident  that  an  increase  of  one  term  must  diminish  tbe 
other.  When  quantities  mutually  limiting  each  other  enter 
into  a  proportion,  it  is  called  Indirect  proportion.  No  special 
rule  however  is  needed  for  the  statement  of  such  questions ; 
for  you  can  always  determine  by  strict  attention,  whether  the 
statement  you  make  is  reasonable. 

29.  If,  with  a  speed  of  11  miles  per  hour,  a  journey  is- 
performed  in  37  hours,  how  long  will  it  take  to  perform  the 
same  journey  with  a  speed  of  15  miles  per  hour  ? 

30.  If  a  stable  keeper  has  grain  for  the  supply  of  29 
horses  43  days,  how  long  will  the  supply  last,  if  he  buys  6 
horses  more  ? 

31.  If .  a  barrel  of  flour  last  a  family  of  7  persons  6  weeks, 
how  long  will  it  last  15  persons  ? 

32.  If  42  men  can  do  a  job  of  work  in  60  days,  how  long 
will  it  take  53  men  to  perform  the  same  work  ? 


ir 


COMPOUND   TROPORTION.  185 

83.  A  ship  builder  employs  50  men  to  complete  a  ship, 
which  they  can  do  in  45  days ;  if  7  of  the  men  fail  to  engage 
in  the  work,  how  long  will  it  take  the  others  to  perform  it  ? 

34.  If  8  yds.  of  cloth,  7  qrs.  wide,  cost  54  dollars,  what 
will  be  the  cost  of  15  yds.  of  cloth,  of  the  same  quality,  9 
qrs.  wide  ? 

Tn  this  example,  the  length  of  the  two  pieces  of  cloth  will 
not  represent  the  ratio  of  their  values,  for  they  are  of  differ- 
ent widths.  The  answer  can  be  found  by  two  statements. 
First,  regarding  the  two  pieces  as  of  the  same  width, 

8  :  15  : :  54  :  lOlJ  dollars,  first  ans. 

JSText,  taking  the  width  into  view, 

7:9::  lOlJ  :  130^\  dollars,  final  ans. 

If  we  examine  the  above  question,  we  shaU  see,  that  in 
neither  of  thein  is  the  quantity  of  cloth  expressed ;  but,  in     ^ 
the  first  statement,  its  length,  and  in  the  second,  its  width. 
Now  the  quantity  of  cloth  is  expressed  by  the  length  multi- 
plied by  the  breadth.     In  the  smaller  piece,  it  is  7X8=56       y 
qrs.  of  a  sq.  yard  ;  in  the  larger  piece,  it  is  15X9^135  qrs.      f 
of  a  sq.  yard.      These  numbers,  56  and   135,  express  the 
quantities  of  cloth ;  and  taking  these,  instead  of  the  dimen- 
sions, a  single  proportion  gives  us  the  answer. 

56:  135  ::  54:  130/^.  -  V', 

As  the  question  is  first  stated,  you  observe,  that,  instead        ^ 
of  the  numbers  which  form  the  ratio,  56  :   135,  you  have 
only  the  factors  of  those  numbers  given.     This  is  called  a 
Compound  Proportion,  * 

A  Compound  Proportion,  then,  is  one  in  which  two  or  more 
of  the  terms  of  the  simple  proportion  are  expressed  in  the  form 
of  their  factors. 

Every  question  containing  a  Compound  Proportion  may 
be  solved  by  means  of  two  or  more  simple  proportions ;  or,  it 
may  be  reduced  to  one  simple  proportion,  as  is  seen  in  Ex. 
34,  above.  This  method,  however,  often  requires  calculations 
in  large  numbers ;  it  may  therefore  be  desirable  to  have  a 
method  given  by  which  the  process  may  be  made  less  tedious. 

The  following  rule  is  offered,  as  applicable  to  all  cases  of 
Proportion,  Simple,  or  Compound,  —  Direct,  or  Inverse.  It  is 
16* 


186  COMPOUND    PROPORTION.  > 

sliort,  and  the  attention  required  in  applying  it  will  afford  a 
good  discipline  for  the  reasoning  powers. 

Mule  of  Proportion. 

Draw  a  horizontal  line ;  then  examine  the  conditions  of 

the  question,  and  consider,  in  the  case  of  each,  whether  its 

increase  would  make  the  answer  greater,  or  smaller ;  if  it 

would  make  it  greater,  set  it  above  the  line  ;  if  smaller,  set  it 

below. 

N  Regard  the  numbers  thus  set  down,  as  the  terms  of  a  com- 

^^^  pound  fraction ;  cancel  common  factors ;  multiply  together  the 

pr\  terms  that  remain,  for  the  answer. 

^'  £ixample. 

k  •»       35.  If  8  men  build  a  wall,  36  ft.  long,  12  ft.  high,  and  4 
J       ft.  thick,  in  72   days,  when  the  days  are  9  hours  long,  how 

many  men  will  build  a  wall  100  ft.  long,  10  ft.  high,  and  3  ft. 

thick,  in  24  days,  when  the  days  are  10  hours  long? 

Cancelling  like  factors  above  and  below  the  line, 
\  Operation,     and  multiplying  the  remaining  terms. 

8  72  9  100  10  3 

.=3L§<i=37i  men.     Answer. 

86  12  4     24  10  ^  ^ 

^  Explanation, 

The   question   is, ,"  how  many  men  ? "     "  If  8   men  will 

\  build,"  &c.  Now,  if  it  took  80  men  to  build  the  first  wall 
instead  of  8,  it  would  require  more  men  to  build  the  second ; 
then  put  8  above  the  line.  "  Build  a  wall  36  ft.  long."  If  it 
were  360  feet  long,  instead  of  36,  it  would  take  fewer  men  to 
build  the  second  wall ;  therefore,  put  36  below  the  line.    Pur- 

*        Bue  the  same  reasoning  with  all  the  other  conditions. 

y  36.  in 5  horses  consume  40  tons  of  hay  in  30  weeks, 

how  many  horses  will  consume  6Q  tons  of  hay  in  32  weeks  ? 

^  37.  If  1  dollar  gain  .06  of  a  dollar  interest,  in  12  months, 

how  much  will  740  dollars  gain  in  8  months  ? 

38.  If  a  crew  of  75  men  have  provisions  for  5  months, 
allowing  each  man  30  oz.  per  day,  what  must  be  the  allow- 
ance per  day,  to  make  the  provisions  last  6^  months  ? 

39.  If  18  brick  layers,  in  12  days,  of  9  hours  each,  build  a 
wall  175  feet  long,  2  feet  thick,  and.  18  feet  high,  in  how 


PROPORTION.  187 

many  days  will  6  men,  working  10  hours  each  day,  build  a 
wall  100  feet  long,  l^-  feet  thick,  and  16  feet  high  ? 

40.  If  10  masons  lay  160  thousand  of  bricks  in  12  days, 
working  8  hours  per  day,  how  many  men  will  lay  224  thou- 
sand in  15  days,  of  10  hours  each? 

Partnership, 

41.  Two  men  trade  in  company ;  one  puts  in  1000  dollars ; 
the  other  2000,  for  the  same  length  of  time ;  they  gain  600 
dollars ;  what  is  each  one's  share  of  the  gain  ? 

It  is  evident  that  each  man's  share  ought  to  bd  in  propor- 
tion to  the  sum  he  put  in.  ' 

As  the  whole  investment  is  to  each  partner's  investment,  so 
is  the  whole  gain  or  loss,  to  each  partner's  gain  or  loss. 

42.  Two  men  trade  in  company  ;  one  puts  in  3500  dollars ; 
the  other  4000  ;  they  gain  600  doUai-s ;  what  is  each  one's 
share  of  the  gain  ? 

43.  Three  men  trade  in  company ;  the  first  puts  in  3400 
dollars;  the  second,  800;  and  the  third,  1200  dollai's;  they 
gain  475  dollars  ;   what  is  each  partner's  share  ? 

44.  Two  men  purchase  a  ship  for  11000  dollars;  one  pays 
8000  dollars; 'the  other  3000;  they  sell  the  ship  for  9500 
dollars  ;  what  was  each  one's  loss  ? 

45.  Two  men  trade  in  company ;  one  puts  in  1000  dollars 
for  6  months,  the  other  puts  in  1000  for  18  months ;  they 
gain  600  dollars  ;  what  was  each  one's  share  of  the  gain  ? 

Here,  though  the  money  was  equal,  it  is  evident  the  gain 
of  one  ought  to  be  three  times  as  great  as  the  other,  because 
his  money  was  in  three  times  as  long. 

Where  the  investments  are  made  for  different  times,  each 
partner's  interest  will  be  expressed  by  multiplying  his  money 
by  the  time  it  was  in  trade.  Then,  as  the  sum  of  all  the  in- 
terests is  to  each  partner's  interest,  so  is  the  whole  gain  or 
loss,  to  each  partner's  gain  or  loss. 

46.  Three  men  trade  in  company ;  the  first  puts  in  400 
dollars  for  8  months;  the  second,  1100  dollars  for  6  months; 
the  third,  1000  dollars  for  7  months ;  they  gain  840  dollars ; 
what  is  each  man's  share  ? 

47.  Four  men  trade  in  company  ;  the  first  puts  in  1200  for 
2  years ;  the  second,  1500  dollars  for  18  months ;  the  third,  600 
for  8  months ;  the  fourth,  900  dollars  for  6^  months ;  they  gain 
1340  dollars ;  what  was  each  man's  share  ? 


188  PROGRESSION. 

SECTION    XL.. 

PROGKESSION. 

When  a  series  of  numbers  is  given,  each  one  of  wliich  has 
the  same  ratio  to  the  number  which  follows  it,  the  series  is 
called  a  progression. 

Progression  is  Arithmetical  or  Geometrical.  Arithmetical 
Progression  is  made  by  the  successive  addition  or  subtraction 
of  a  common  difference.  When  the  common  difference  is 
added  to  each  term  in  order  to  make  the  succeeding  one,  the 
series  is  called  an  ascending  series;  as  1,  3,  5,  7,  9,  11,  &c. 

When  the  common  difference  is  subtracted,  the  series  is 
called  a  descending  series;  as  11,  9,  7,  5,  3,  1. 

If  you  know  the  first  term,  and  the  common  difference  of 
an  Arithmetical  Progression,  you  can  write  the  whole  series, 
for  to  do  this,  you  have  only  to  add  or  subtract  the  common 
difference  for  each  succeeding  term.  If  the  whole  series  is 
written  out,  it  is  evident  you  can  find  by  inspection,  any  par- 
ticular term,  as  the  7th,  the  15  th,  the  20th,  &c.  But,  if  the 
series  be  a  long  one,  this  may  be  a  very  tedious  operation. 

Suppose  the  series  given  above,  1,  3,  5,  7,  &c.,  were  con- 
tinued to  87  terms,  and  you  were  required  to  find  what  was 
the  last  term. 

By  examining  the  series,  you  will  see  that  the  2d  term=the 
Ist-f-the  common  difference ;  the  3d=lst-[-twice  the  com- 
mon difference ;  the  4th=lst-|-3  times  the  common  differ- 
ence; the  5th=lst-|-4  times  the  common  difference,  &c. 
Any  term  whatever  equals  the  lU  term,~\-the  common  difference, 
multiplied  by  a  number  one  less  than  that  which  expresses  the 
place  of  the  term.  The  87th  term  in  the  above  series,  there- 
fore, is  1+2X86=173. 

1.  What  is  the  38th  term  of  the  series  1,  3,  5,  7,  &c.  ? 

2.  What  is  the  53d  term  of  the  same  series  ?  the  91st 
term?  the  89th  term  ?  the  107th  term? 

3.  In  an  arithmetical  series,  the  first  term  of  which  is  1, 
and  the  common  differrnce  3,  what  is  the  64th  term  ?  the 
75th  term  ?  the  81st  term  ? 

4.  In  the  series  1,  5,  9,  13,  &c.,  what  is  the  40th  term?  the 
67th  term?  the  80th  term  ? 


ARITHMETICAL    PROGRESSION.  189 

5.  In  the  series  2,  4,  6,  &c.,  what  is  the  45th  term  ?  What 
is  the  lOQth  term  ?     What  is  the  200th  term  ? 

Hence,  if  you  know  the  number  and  place  of  any  term,  and 
the  common  difference,  you  may  find  the  first  or  any  other  term. 

6.  If  the  5th  term  of  an  arithmetical  series  is  13,  and  the 
common  difference  3,  what  is  the  1st  term  ?  What  is  the 
24th  term  ?     What  is  the  191st  term  ? 

7.  If  the  6th  term  of  a  series  is  77,  and  the  common  differ- 
ence 15,  what  is  the  2d  term  ?     What  is  the  14th  term  ? 

8.  If  the  22d  term  in  a  series  is  89,  and  the  common  differ- 
ence 4,  what  is  the  10th  term  ?     What  is  the  43d  term  ? 

By  knowing  the  number  and  the  place  of  any  two  terms,  we 
may  find  the  common  difference. 

9.  In  a  certain  series  the  4th  term  is  10,  and  the  7th  term 
is  19  ;  what  is  the  common  difference  ? 

19 — 10=9  ;  now  this  difference,  9,  is  made  by  the  addition 
of  the  common  difference  three  times  ;  for  7 — 4=3 ;  the  com- 
mon difference,  therefore,  is  9-^3=3. 

10.  In  a  certain  series  the  5th  term  is  9,  and  the  1 1th  term 
is  21 ;  what  is  the  common  difference  ? 

11.  In  a  certain  series  the  4th  term  is  13,  and  the  9th  term 
is  33  ;  what  is  the  common  difference  ? 

If  we  know  the  1st  term,  the  common  difference,  and  the 
number  of  terms,  we  can  find  the  sum  of  all  the  terms. 

12.  How  many  strokes  does  a  clock  strike  in  24  hours,  from 
noon  to  noon  ? 

We  might  write  down  the  series,  1,  2,  3,  &c.,  up  to  12, 
which  would  express  the  number  of  strokes  in  12  hours,  from 
noon  till  midnight ;  we  might  write  the  same  series  again,  for 
the  time  from  midnight  tiU  noon  ;  and  by  adding  these  num- 
bers together,  might  obtain  the  answer.  But  a  much  shorter 
way  may  be  found.  To  exhibit  it  we  wdll  write  the  two  series 
thus  : 

1st  series,     12345678     9  10  11  12  from  noon  tm  midnight. 
2d  series,   121110     9     8     7     6     5     4     3     2     1  from  midniglit  tiU  noon. 

13  13  13  13  13  13  13  13  13  13  13  13 
Sum  of  both  series  equal  to  12  times  13.  But  13  is  the 
sum  of  the  first  and  last  terms  ;  and  12  is  the  number  of  terms. 
Therefore,  the  sum  of  the  first  and  last  terms,  multiplied  by  the 
number  of  terms,  gives  the  sum  of  all  the  terms  of  both  series 
Half  this  number  will  be  the  sum  of  one  series. 


190  ARITHMETICAL    PROGRESSION. 

13.  What  is  the  sum  of  the  series  1,  4,  7,  to  20  terms  ? 
First  find  the  20th  term. 

14.  What  is  the  sum  of  50  terms  of  the  series  2,  G,  10  ? 

15.  A  farmer  instructed  his  boy  to  carry  fencing-posts  from 
a  pile  to  the  holes  in  the  ground  where  they  were  to  be  in- 
serted, taking  one  post  at  a  time ;  the  holes  are  1 2  feet  apart, 
in  a  straight  line,  and  the  pile  of  posts  30  feet  from  the  first 
hole ;  how  far  must  he  travel,  in  carrying  to  their  places  100 
posts  ? 

1 6.  If  the  hours  in  a  whole  week  were  numbered  in  regular 
progression,  and  were  struck  in  this  way  by  the  clock,  how 
many  strokes  must  the  clock  strike  for  the  last  hour  of  the 
week  ? 

What  would  be  the  whole  number  of  strokes  in  the  week  ? 
If  we  know  the  first  and  last  terms  and*  the  common  differ- 
ence, we  can  find  the  number  of  terms. 

17.  The  first  term  of  a  series  is  4;  the  last  term  is  19;  the 
common  difference  is  3  ;  what  is  the  number  of  terms  ? 

The  difference  between  the  extremes  19 — 4,  is  15  ;  this, 
you  know,  is  the  common  difference  3  ;  taken  a  certain  number 
of  times ;  15-i-3=5  ;  there  are  then  5  additions  of  the  common 
difference  ;  now  the  number  of  terms  is  1  more  than  the  num- 
ber of  times  the  common  difference  has  been  added.  To  find 
the  number  of  terms,  then 

Find  the  difference  of  the  extremes  ;  divide  it  hy  the  common 
difference  ;  increase  the  quotient  hy  Ifor  the  number  of  terms, 

DESCENT  OF  FALLING  BODIES. 

18.- A  body  falling  through  the  air  falls,  the  1st  second, 
16.1  feet  ;*  in  the  2d  second,  48.3  feet ;  in  the  3d  second,  80.5 
feet ;  how  many  feet  farther  does  it  fall  each  second  than  it 
fell  the  second  before  ? 

19.  Taking  the  answer  to  the  preceding  question  as  the 
common  difference,  and  16.1  as  the  first  term  of  a  series,  how 
far  will  a  body  fall  in  4  seconds  ? 

^  This  is  a  more  exact  statement  than  that  made  in  Part  I.  See  Olm- 
Btcad's  Natural  Philosophy.  It  should  be  remarked,  also,  that  no  allow- 
ance, in  these  examples,  is  made  for  the  resistance  of  the  atmosphere, 
which  always  diminishes  the  speed  somewhat,  and  becomes  greater  and 
greater  as  the  speed  increases. 


GEOMETRICAL   PROGRESSION.  191 

20.  How  far  will  a  body  fall  in  5  seconds  ? 

21.  How  far  will  a  body  fall  in  6  seconds? 

22.  A  stone  in  falling  to  the  ground,  falls  the  last  second 
209.3  feet ;  how  many  seconds  has  it  fallen ;  and  from  what 
height? 

23.  A  stone  in  falling  to  the  ground,  falls  the  last  second 
241.5  'feet ;  how  many  seconds  has  it  fallen ;  and  from  what 
height  ? 

24.  If  a  stone  dropped  into  a  well  strikes  the  water  in  3 
seconds,  how  far  is  it  to  the  surface  of  the  water  ? 


SECTION    XLI. 

GEOMETKICAL  PROGEESSION. 

A  series  of  numbers  such  that  each  is  the  same  part  or  the 
same  multiple  of  the  number  that  follows  it,  is  called  a  geomet- 
rical series.  The  ascending  series,  1,  3,  9,  27,  is  of  this  kind, 
for  each  term  is  one  third  of  that  which  succeeds  it.  So,  in 
the  descending  series,  64,  16,  4,  1,  each  term  is  4  times  the 
following  term. 

The  number  obtained  by  dividing  any  term  by  the  term 
before  it,  is  called  the  ratio  of  the  progression.  Thus,  in  the 
first  of  the  above  examples,  the  ratio  is  3  ;  in  the  second  exam- 
ple it  is  :^. 

Let  us  take  the  series,  2,  6,  18,  54,  and  observe  by  what 
law  it  is  formed.  The  ratio  is  3 ;  the  first  term,  2.  The 
second  term  is  2X3,  or  the  first  term  X  the  ratio  ;  the  third 
term  is  2X3^,  o;  the  first  term  X  the  second  power  of  the 
ratio  ;  the  fourth  term  is  2X3^,  or  the  flrst  term  X  the  third 
power  of  the  ratio. 

Thus  each  term  consists  of  the  first  term  multiplied  hy  the 
ratio,  raised  to  a  power  whose  index  is  one  less  than  the  number 
expressing  the  place  of  the  term. 

1.  What  is  the  7th  term  in  the  series  1,  4,  16,  &c.? 

2.  What  is  the  10th  term  in  the  series  3,  6,  12,  &c.? 

3.  A  glazier  agrees  to  insert  a  window  of  16  lights  for  what 
the  last  light  would  come  to,  allowing  1  cent  for  the  first  light, 
2  for  the  second,  and  so  on ;  what  did  the  wiudow  cost  ? 


^  Wg 


192  MENSURATION  OF  SURFACES. 

4.  If,  in  tlie  year  1850,  the  population  of  the  United  States 
shall  be  20000000,  and  if  it  shall  thenceforward  double  once 
in  every  thirty  years,  what  would  be  the  population  in  1970  ? 

To  obtain  the  sum  of  the  terms,  when  the  first  and  last 
terms  are  given,  and  the  ratio, 

Rule.  —  Multiply  the  last  term  by  the  ratio,  suhtract^the  first 
term  from  this  product,  and  divide  the  remainder  hy  the  ratio 
diminished  hy  one. 

5.  A  gentleman  promises  his  son,  11  years  old,  one  miU 
when  he  shall  be  12  years  old,  and,  on  each  succeeding  birth- 
day till  he  is  21  years  old,  ten  times  as  much  as  on  the  pre- 
ceding birth-day ;  what  will  the  son's  fortune  be,  without 
interest,  when  he  is  21  years  old  ? 


SECTION    XLII. 

MENSURATION   OF  SURFACES. 

For  the  mensuration  of  the  triangle  and  the  parallelogram, 
when  the  base  and  height  are  known,  see  Sec.  XVII.  Part  I. 

To  find  the  area  of  an  equilateral  triangle  when  the  sides 
only  are  known. 

Square  one  side  ;  multiply  that  product  hy  the  decimal  .433. 

To  find  the  circumference  of  a  circle,  when  the  diameter  is 
.known. 

Multiply  the  diameter  hy  3.1416. 

1.  What  is  thejcircumference  of  a  circle  the  diameter  of, 
which  is  36  feet^'i?^ 

2.  What  is  the  circumference  of  a  circular  race-course 
whose  diameter  is  1;^  miles  ? 

3.  What  is  the  circumference  of  a  wheel  the  diameter  of 
which  is  24  feet,  6  inches  ? 

4.  What  is  the  <^^^^mference  of  the  earth  on  the  line  of  the 
equator,  its  diameter  being  7925.65  miles  ? 

To  find  the  dia^ieter  of  a  circle,  when  the  circumference  is 
known. 

Divide  the  circumference  Jy  3.1416. 


MENSURATION   OP    SOLIDS.  193 

5.  How  far  is  it  across  a  circular  pond,  the  circumference 
of  which  is  231  rods? 

To  find  the  area  of  a  circle,  when  the  diameter  and  circum- 
ference are  known, 

Multiply  the  circumference  hy  one  fourth  of  the  diameter;  or, 
multiply  the  square  of  the  diameter  by  the  decimal  .7854. 

6.  What  is  the  area  of  a  circle  whose  diameter  is  34  rods  ? 

7.  What  is  the  area  of  a  circle  whose  diameter  is  -24  feet  ? 
When  a  circle  is  given,  to  find>a  square,  which  shall  have 

an  equal  area, 

Find  the  area  of  the  circle,  extract  the  square  root,  which 
will  be  one  side  of  the  square, 

8.  There  is  a  circular  piece  of  land,  40  rods  in  diameter ; 
what  will  be  the  side  of  a  square  of  equal  area  ? 

9.  There  is  a.  circular  green,  containing  8  acres ;  what  will 
be  the  side  of  a  square  of  equal  area  ? 

10.  There  is  a  circle  35  rods  in  diameter,  and  a  square  31 
rods  ;  which  is  the  greater ;  and  how  much  ? 


SECTION    XLIII. 

MENSUKATION    OF    SOLIDS. 

A  plane  solid  is  bounded  by  flat  surfaces ;  a  round  solid  is 
bounded  by  curved  siu-taces. 

To  find  the  surface  of  a  solid  bounded  by  plane  surfaces, 

Find  the  area  of  each  plane  surface,  and  add  the  sums  to- 
gether for  the  whole  surface, 

A  prism  is  a  solid,  whose  ends  are  any  equal,  parallel,  and 
similar  rectileneal  figures,  and  whose  sides  are  parallelograms. 

To  find  the  solidity  of  a  prism, 

Multiply  the  area  of  the  base  by  the  height. 

1.  What  is  the  solidity  of  a  prism  whose  ends  are  equilat- 
eral triangles,  14  inches  on  a  side,  and  whose  height  is  8  feet  ? 

A  cylinder  is  a  round  solid,  whose  ends  are  equal, and 
parallel  circles. 

To  find  the  solidity  —  the  same  ride  as  for  a  prism, 
17 


194  MENSURATION    OF   SOLIDS. 

2.  "What  is  the  solid  contents  of  a  cylinder,  whose  ends  are 
circles  18  inches  in  diameter,  and  whose  height  is  12  feet? 

A  regular  pyramid  is  a  solid,  whose  sides  are  equal  and 
similar  triangles,  meeting  in  a  point  at  the  top.  The  slant- 
height  is  the  distance  from  the  point,  at  the  top,  to  the  middle 
of  the  base  of  one  of  the  triangles. 

To  find  the  solid  contents  of  a  pyramid, 

Multiply  the  area  of  the  base  by  vne  third  the  peiyendicular 
height. 

3.  What  is  the  solid  contents  of  a  four-sided  pyramid,  whose 
J)ase  measures  40  feet  on  a  side,  and  whose  height  is  42  feet  ? 

A  cone  is  a  round  solid,  standing  on  a  circular  base,  and 
terminating  in  a  point  at  the  top. 

To  find  the  solidity  of  a  cone  —  the  same  rule  a^  for  a 
pyramid. 

4.  What  is  the  solidity  of  a  cone,  whose  base  is  13  feet  in 
diameter,  and  whose  height  is  22  feet  ? 

5.  What  is  the  solidity  of  a  cone,  whose  base  is  4  feet  in 
diameter,  and  height  18  feet  ? 

To  find  the  surface  of  a  sphere, 

Multiply  the  diameter  by  the  circumference. 

6.  How  many  square  miles  of  surface  has  the  earth,  regard- 
ing it  as  a  sphere  the  diameter  of  which  is  7925.65  miles  ? 

To  find  the  solidity  of  a  sphere. 

Multiply  the  surface  by  |  o/  the  diameter  ;  or,  multiply  the 
cube  of  the  diameter  by  the  decimal  .5236. 

To  find  the  measure  of  a  sphere,  when  the  solidity  is  given, 

Divide  the  solidity  by  the  decimal  .5236 ;  extract  the  cube 
root  of  the  quotient,  which  will  be  the  diameter  of  the  sphere. 

7.  What  is  the  solid  contents  of  a  sphere  the  diameter  of 
which  is  14  inches  ? 

8.  What  will  be  the  solidity  of  the  largest  sphere  that  can 
be  cut  from  a  cubic  block  1  foot  on  a  side  ? 


MISCELLANEOUS   THEOREMS   AND    QUESTIONS.        195 


SECTION    XLIV. 

MISCELLANEOUS  THEOREMS  AND  QUESTIONS. 

Given  the  sum  and  the  difference  of  two  numbers,  to  find 
the  larger  and  the  smaller  numbers, 

Add  half  the  difference  to  half  the  sum,  for  the  larger ;  sub- 
tract half  the  dfference  from  half  the  sum,  for  the  smaller. 

1.  The  sum  of  two  numbers  is  140,  the  difference  32;  what 
are  the  numbers  ? 

2.  The  sum  of  two  numbers  is  572,  the  difference  94;  what 
are  the  numbers  ? 

3.  The  sum  of  two  numbers  is  187,  the  difference  44;  what 
are  the  numbers  ? 

4.  The  sum  of  two  numbers  is  190,  the  difference  57;  what 
are  the  numbers  ? 

Given  the  sum  and  the  product  of  two  numbers,  to  find  the 
larger  and  the  smaller  number. 

5.  There  are  two  numbers ;  their  sum  is  80,  and  their  pro- 
duct 1551 ;  what  are  the  numbers  ? 

The  theorem  on  which  the  solution  of  this  question  depends 
is  this,  If  a  number  be  divided  into  two  equal  parts,  and  also 
into  two  unequal  parts,  the  product  of  the  two  equal  parts, 
that  is,  the  square  of  half  the  number,  will  equal  the  product 
of  the  two  unequal  parts,  plus  the  square  of  the  difference 
between  one  of  the  equal  and  "one  of  the  unequal  parts. 

Take  16;  divide  it  equally,  8-1-8,  and  unequally,  9-|-7  ; 
the  difference  between  an  equal  and  an  unequal  part,  1 ;  82= 
64  :  9X7=63  ;  loss  1,  which  is  the  square  of  the  difference. 

Divide  unequally  into  10-|-6  ;  difference  2  ;  82=64  :  lOX 
6=60 ;  loss  4,  which  is  the  square  of  the  difference. 

Divide  unequally  into  ll-f-5  ;  difference  3  ;  8^=64  :  11 X 
5=55 ;  loss  9,  which  is  the  square  of  the  difference. 

Hence,  to  solve  the  question,  —  subtract  the  product  of  the 
unequal  parts  from  the  square  of  half  the  number,  find  the 
square  root  of  the  difference,  add  it  to  half  the  number  for  the 
greater,  sttbtract  it  from  half  the  number  for  the  less. 

6.  There  are  two  numbers,  the  sum  of  which  is  100 ;  their 
product  is  2419  ;  what  are  the  numbers  ? 


196  MISCELLANEOUS    QUESTIONS. 

.  7.  There  is  a  rectangular  piece  of  land,  the  two  contiguous 
boundaries  of  which  measure  together  120  rods  ;  the  area  of 
the  piece  is  2975  square  rods ;  what  is  it§  length  ?  What  is 
its  width  ? 

8.  A  rectangular  piece  of  land  is  surrounded  by  480  rods 
of  fence ;  the  area  is  13104  square  rods  ;  what  is  its  length 
and  breadth  ? 

Given  the  sura  of  two  numbers,  and  the  difference  of  their 
squares,  to  find  the  greater  and  the  less  number, 

The  'product  of  the  sum  and  the  difference  of  two  numbers 
is  equal  to  the  difference  of  their  squares. 

Take  the  two  numbers,  6  and  9 ;  their  sum  is  15  ;  their  dif- 
ference 3  ;  15  X  3=45.  The  square  of  6  is  36 ;  the  square  of 
9  is  81 ;  81—36=45. 

Hence,  if  we  divide  the  difference  of  the  squares  by  the 
sum,  the  quotient  :^'ill  be  the  difference,  and  from  this  we  may 
find  the  gi-eater  and  the  less. 

9.  There  is  a  triangle,  the  hypotenuse  and  one  leg  of  which 
measure  together  90  feet ;  the  other  side  measures  40  feet ; 
what  are  the  lengths  of  the  two  first-named  sides  respectively  ? 

10.  There  is  a  triangle  ;  the  hypotenuse  and  base  measure 
together  120  feet ;  the  perpendicular  measures  64  feet ;  what 
is  the  length  of  each  of  the  first-named  sides  ? 

This  principle  enables  you  to  multiply  readily  any  two 
numbers,  one  of  which  exceeds  a  certain  number  of  tens  by 
as  many  units  as  the  other  falls  short  of  it ;  as  63  X-57  ;  the 
first  exceeds  60  by  3  ;  the  second  falls  short  of  it  by  3. 

Square  the  tens, — 3600 ;  subtract  from  this  the  square  of 
the  units — 9  ;  3591,  answer. 

11.  Multiply  64X56,  3600—16=3584,  answer. 

12.  Multiply  82X78;  47X33;  92X88. 

Theorem  of  Parallel  Sections. 

If  a  line  is  drawn  in  a  triangle,  parallel  to  one  of  the  .sides, 
and  meeting  the  other  two  sides,  it  divides  those  sides  propor- 
tionally ;  and  the  small  triangle  cut  off,  is  similar  to  the  whole 
undivided  triangle. 

If  a  plane  pass  through  a  pyramid  or  cone,  parallel  to  the 
base,  it  divides  allUhe  lines  it  meets  proportionally;  and  the 
siTiall  solid  cut  off  at  the  top  is  similar  to  the  whole  undivided 
{ Dlid. 


MISCELLANEOUS   QUESTIONS.  197 

13.  There  is  a  triangular  field,  containing  7  acres ;  a  line  is 
drawn  through  it,  parallel  to  one  side,  cutting  the  other  two 
sides  f  of  the  distance  from  the  apex  to  the  base ;  how  much 
land  does  it  cut  off? 

14.  There  is  a  board  in  the  form  of  a  right  angled  triangle, 
8  feet  in  perpendicular  height ;  how  far  from  the  top  must  a 
line  be  drawn  parallel  to  the  base  to  cut  off  f  of  the  board  ? 

15.  A  man  had  a  field  of  3  acres,  in  the  form  of  a  right 
angled  triangle,  with  the  base  equal  to  the  perpendicular ;  he 
sells  one  acre,  to  be  cut  off  bj  a  line  running  parallel  to  the 
base  ;  he  then  sells  another  acre,  to  be  cut  off  by  another  line 
parallel  to  the  base ;  how  far  from  the  base  must  the  first  line 
be  ?     How  far  from  the  base  must  the  second  line  be  ? 

16.  There  was  a  cone,  20  feet  high;  but  the  upper  part 
being  defective,  11  feet  in  height  of  the  top  was  taken  down ; 
how  much  of  the  cone  has  been  removed  ? 

17.  There  was  a  square  pyramid,  the  base  of  which  meas- 
ured 48  feet  on  a  side ;  when  it  was  partly  completed,  its  slant 
height,  measuring  from  the  middle  of  a  side  at  the  bottom  to 
the  middle  of  the  same  side  at  the  top,  was  40  feet,  and  the 
width  of  a  side  at  the  top  was  12  feet;  how  high  was  the  apex 
of  the  pyramid,  when  completed  ?  and  what  part  of  the  pyra- 
mid remained  to  be  built  ? 

18.  In  a  right  angled  triangle,  whose  base  and  perpendicu- 
lar are  equal,  what  is  the  ratio  of  the  square  of  the  hypotenuse 
to  the  square  of  the  base  ?  What  is  the  ratio  of  the  hypote- 
nuse to  the  base  ? 

19.  If  a  man  travel  on  Monday  6  miles,  due  north,  and  on 
Tuesday  8  miles,  due  east ;  how  far  is  he  then  from  where  he 
set  out  on  Monday  ? 

If  on  Wednesday  he  travels  12  miles,  due  south,  how  far 
will  he  then  be  from  w^here  he  was  on  Monday  morning  ? 
How  far  from  where  he  was  on  Monday  night  ? 

20.  In  repairing  a  meeting-house,  it  Avas  thought  desirable 
to  alter  the  form  of  the  posts,  which  were  one  foot  square. 
It  was  proposed  to  cut  away  the  corners,  so  as  to  make  them 
regular  eight-sided  prisms.  HoW  wide  must  each  face  be,  so 
as  to  have  all  the  eight  faces  of  exactly  the  same  width  ? 

21.  Sound  moves  through  the  air  at  the  rate  of  1090  feet 
in  a  second  ;  how  long  would  it  be  in  passing  100  miles  ? 

22.  At  the  above  rate,  how  long  would  it  require  for  a 
wave  of  sound  to  compass  one  half  the  circuit  of  the  globe, 

17* 


198  SPECIFIC    GRAVITY. 

on  the  line  of  the  equator,  the  circumference  being  24899 
miles  ? 

23.  Two  men  purchase  in  equal  shares,  a  stick  of  hewn 
timber,  40  feet  long,  2  feet  square  at  the  larger  end,  and  1 
foot  square  at  the  smaller  end ;  how  far  from  the  larger  end 
shall  they  cut  it  in  two,  so  that  each  may  have  exactly  one 
half? 

24.  A  surveyor,  in  laying  out  a  lot  of  land,  first  runs  a 
line  due  North,  to  a  certain  tree ;  from  the  tree  he  runs  be- 
tween South  and  West  till  he  comes  to  a  point  due  West 
from  the  place  he  started  from ;  the  whole  of  these  two  lines 
is  212  rods,  but  those  who  measured  it  neglected  to  note  how 
far  the  tree  was  from  the  starting  point ;  on  measuring  a 
third  line,  connecting  the  extremities  of  the  two  first  lines, 
they  find  it  98  rods ;  how  many  acres  does  the  triangle  con- 
tain ? 

Specific  Gravity. 

The  specific  gravity  of  a  body,  is  its  weight  compared  with 
the  weight  of  an  equal  bulk  of  water.  To  find  the  specific 
gravity  of  a  body  heavier  than  water. 

Weigh  the  body  in  water,  and  out  of  water,  and  find  the 
difference  in  the  weight ;  then,  as  the  difference  in  the  weight 
is  to  the  weight  out  of  water,  so  is  1  to  the  specific  gravity. 

The  weight  of  a  cubic  foot  of  water  is  62|^  lbs.  av.  The 
specific  gravity  of  the. most  important  of  the  metals  is  as 
follows : 

Iron,     7.78  Tin,     .  7.2  Copper,  8.895 

Lead,  11.325         Mercury,    13.568         Silver,  10.51 
Gold,  19.257         Platinum,    21.25 

From  the  above  table,  we  may  find  the  weight  of  any  mass 
dF  one  of  the  above  metals  the  magnitude  of  which  is  known. 

25.  What  is  the  weight  of  a  cubic  foot  of  iron  ? 

26.  What  is  the  weight  of  an  iron  ball  six  inches  in  diam- 
eter? 

27.  What  is  the  diameter  of  a  24  lb.  cannon  ball  ? 

28.  What  is  the  diameter  of  a  48  lb.  cannon  ball  ? 

29.  What  is  the  weight  of  a  cannon  ball  one  foot  in  di- 
ameter ?    . 

30.  If  the  column  of  mercury  in  the  barometer  be  29^ 


MECHANICAL   POWERS.  199 

inches  liigh,  what  would  be  the  weight  of  a  column  of  mer- 
cury of  that  height,  one  inch  square  ? 

31.  As  the  weight  of  mercury  in  the  barometer  equals  the 
weight  of  the  atmosphere  on  the  same  base,  what  is  the  pres- 
sure of  the  atmosphere  on  a  square  foot,  when  the  mercury 
in  the  barometer  is  29  inches  high  ? 

The  height  to  which  water  will  rise  in  a  suction  pump, 
and  the  height  of  the  mercury  in  the  barometer,  are  in  in- 
verse proportion  to  the  specific  weight  of  those  two  bodies ; 
that  is,  the  water  is  as  much  higher  than  the  mercury,  as 
mercury  is  heavier  than  water. 

32.  How  high  will  water  rise  in  a  suction  pump,  when  the 
mercury  in  the  barometer  is  29^  inches  high  ? 

33.  What  is  the  weight  of  a  copper  prism,  its  base  being 
an  equilateral  triangle,  3  inches  on  a  side,  and  its  height  15 
inches  ? 

Mechanical  Powers. 

The  object  to  be  gained  by  the  application  of  mechanical 
powers,  is  to  overcome  a  large  weight  or  resistance,  by  means 
of  a  comparatively  small  .power. 

In  doing  this,  however,  the  power  must  move  through  a 
space  as  much  larger  than  the  space  which  the  weight  moves 
through,  as  the  weight  is  heavier  than  the  power. 

Or,  the  distancey^weightf  of  the  power^==distance'X.weight, 
of  the  weight,  or  mass  to  be  moved. 

This  is  the  great 'law  of  mechanical  powers,  and  applies 
to  them  all,  without  exception.  In  the  practical  application 
of  them,  a  certain  allowance  must  be  made  on  account  of 
the  friction  in  the  machine.  The  amount  of  friction  differs 
in  different  powers.  No  account  of  this  will  be  taken  in  the 
examples  which  follow,  unless  it  is  particularly  mentioned  ; 
nor  will  any  difference  be  made  between  the  power  when  in 
motion,  and  when  in  equilibrium,  or  at  rest. 

The  Lever. 

The  lever  is  a  straight  bar  used  to  support  or  raise  heavy 
weights.  It  is  supported  by  a  jt)rc»p  or  fulcrum,  placed  near 
the  weight,  and  the  power  is  applied  at  the  other  end  of  the 
lever.    The  distance  from  the  fulcrum  to  the  weight,  is  called 


200  THE    LEVER. 

the  shorter  arm ;  the  distance  from  the  fulcinim  to  the  power, 
the  longer  arm  of  the  lever.  If  the  lever  were  to  turn  over 
the  fulcrum  as  a  centre,  the  longer  arm  would  describe  a 
larger  circle,  and  the  shorter  arm  would  describe  a  smaller 
circle.  The  circumferences  of  these  two  circles,  or  an  arc 
of  the  same  number  of  degrees  in  both,  would  be  the  dis- 
tances passed  through  by  the  power  and  the  weight  respec- 
tively. But  we  may  take  the  arms  themselves  as  represent- 
ing these  distances,  for  they  are  the  radii  of  the  two  circles ; 
and  the  radii  of  different  circles  have  the  same  ratio  to  each 
other  as  the  circumferences. 

We  have  therefore  this  proportion : 

The  longer  krm  is  to  the  shorter  arm,  as  the  weight  to  the 
power.  Or,  let  1,  a.  stand  for  the  longer  arm ;  S.  a.,  for  the 
shorter ;  w.  for  the  weight,  and  p.  for  the  power. 

1.  a.  :  S.  a.  : :  W.  :  p.  ;  and  any  change  admissible  in  the 
terms  of  a  proportion,  may  be  made  in  these  terms. 

34.  If  a  lever  10  feet  long  have  its  fulcrum  one  foot  from 
the  weight,  how  great  must  the  power  be,  to  raise  a  weight 
of  1640  lbs.? 

35.  If  a  lever  IQ  feet  long  have  its  fulcrum  18  inches 
from  the  weight,  how  great  a  weight  will  be  raised  by  a  pow- 
er of  160  lbs.? 

36.  A  lever  18  feet  long  rests  on  a  fulcrum  2  feet  from  the 
end  ;  how  large  a  weight  can  two  m*en  raise,  —  one  weighing 
164  lbs.,  the  other  172  lbs., —  by  applying  their  weight  at  the 
longer  arm  ? 

37.  If  a  lever  7\  feet  long  rest  on  a  fulcrum  15  inches 
from  the  end,  how  heavy  must  the  power  be  to  support  a  ton, 
gross  weight  ? 

38.  If  the  weight  be  3600  lbs.,  and  the  power  140  lbs., 
how  far  from  the  weight  must  the  fulcrum  be  placed  under  a 
lever  12  feet  long,  so  as  to  have  the  weight  and  power  bal- 
ance? 

39.  If  the^  weight  be  6480  lbs.,  the  power  312,  and  the 
lever  16^  feet  long,  how  far  from  the  weight  must  the  ful- 
crum be,  to  have  the  weight  and  power  balance  ? 

40.  In  a  certain  machine,  it  is  necessary  to  adjust  a  lever 
3  feet  long,  so  that  a  power  of  1^  lbs.  shall  balance  13^ 
lbs. ;  how  far  from  the  weight  must  the  fulcrum  be  placed  ? 


THE   WHEEL   AND   AXLE.  201 


The   Wheel  and  Axle. 

In  this  case  the  power  is  applied  at  the  circumference  of 
the  wheel,  and  the  weight  is  drawn  up  by  a  rope  passing 
round  the  axle,  which  is  a  smaller  wheeh  The  principle, 
therefere,  is  the  same  as  in  the  lever ;  the  semi-diameter  of 
Ihe  wheel  is  the  longer  arm ;  the  semi-diameter  of  the  axle, 
ihe  shorter  arm. 

41.  In  a  grocery  store  the  wheel  and  axle  used  in  raising 
heavy  articles,  are  of  the  following  dimensions,  viz. :  the  wheel 
5  feet  in  diameter  the  axle  7  inches  in  diameter ;  what  pow- 
er must  be  applied  to  the  rope  passing  over  the  wheel,  to 
balance  a  barrel  of  flour  weighing  205  lbs.,  suspended  by  a 
rope  passing  over  the  axle  ? 

42.  With  the  same  wheel  and  axle,  what  power  will  raise 
a  box  of  sugar  weighing  431  lbs.,  adding  ^  to  the^  power,  to 
overcome  the  friction  ? 

43.  In  digging  a  well,  the  wheel  employed  in  raising  stones 
and  earth,  is  6  feet  in  diameter ;  the  axle  6^  in.  in  diam.  \  what 
power  will  raise  a  rock  weighing  640  lbs.,  adding  ^  to  the 
power,  to  overcome  the  friction,? 

44.  If  a  wheel  is  14  feet  in  diameter,  what  must  be  the 
diameter  of  the  axle,  in  order  that  a  power  of  140  lbs.  may 
balance  5760  lbs.  ? 

45.  If  an  axle  is  16|  inches  in  diameter,  what  must  be  the 
diameter  of  the  wheel  in  order  that  a  power  of  56  lbs.,  may 
balance  a  weight  of  1344  lbs.  ? 

The  Screw. 

In  this  case,  the  distance  passed  through  by  the  power  in 
one  revolution,  is  equal  to  the  circumference  of  the  circle 
described  by  the  lever  which  turns  the  screw ;  the  distance 
passed  by  the  weight,  is  the  distance  between  two  threads  of 
the  screw,  measured  in  the  direction  of  its  axis. 

In  the  practical  application  of  this  power,  a  large  allow- 
ance must  be  made  to  compensate  for  the  friction. 

46.  If  the  lever  of  a  screw  is  11  feet  in  length,  and  the 
distance  of  the  threads  1^  inches,  what  power  will  raise  a 
weight  of  6431  lbs.,  making  no  allowance  for  friction  ? 


202         STRENGTH   OF   BEAMS   TO   RESIST   FRACTURE. 

47.  With  the  same  conditions  as  in  the  last  example,  what 
weight  will  be  raised  by  a  power  of  1 24  lbs.  ? 

48.  What  must  be  the  length  of  the  lever  of  a  screw- 
the  threads  of  which  are  1  inch  asunder,  in  -order  that  a  pow- 
er of  3  lbs.  may  balance  a  weight  of  1640  lbs.,  making  no 
allowance  for  friction  ? 

49.  How  far  asunder  must  the  threads  of  a  screw  be,  so 
that,  with  a  lever  of  8^  feet  in  length,  26  lbs.  will  balance 
6590  lbs.  ? 

Strength  of  Beams  to  resist  Fracture. 

[See  Section  XX.  Part  I.] 

In  addition  to  the  principles  that  have  already  been  stated 
in  estimating  the  strength  of  timbers,  the  following  are  among 
the  most  important.  It  is  understood  in  all  cases,  when  tim- 
bers are  compared,  that  they  are  of  the  same  wood,  and 
equally  good  in  quality. 

When  the  depth  of  two  beams  is  the  same,  and  the  thick- 
ness the  same,  the  strength  is  inversely  as  the  length. 

50.  There  are  two  beams  of  the  same  depth  and  thickness ; 
one  18  feet  in  length,  the  other  13  ;  the  longer  beam  will 
sustain  a  weight  of  68  cwt. ;  what  weight  will  the  shorter 
beam  sustain  ? 

51.  Two  beams  of  the  same  size,  measure  in  length  22  and 
17^  feet ;  the  shorter  beam  will  sustain  76  cwt. ;  how  much 
will  the  longer  beam  sustain  ? 

52.  Two  beams  of  equal  thickness  have  a  depth  of  14  and 
16  inches  respectively;  the  deeper  beam  is  20  feet  long, 
and  will  sustain  84  cwt. ;  the  other  is  17  feet  in  length ;  what 
weight  will  it  sustain  ? 

First  take  into  view  the  length ;  then,  in  a  second  propor- 
tion, the  depth. 

53.  If  a  beam  25  feet  in  length  and  9  in.  in  depth,  will  sus- 
tain a  weight  of  12  cwt.,  what  weight  will  be  sustained  by  a 
beam  of  the  same  thickness  18  feet  long,  and  10  in.  in  depth  ? 

When  beatns  are  of  the  same  length  and  depth,  the  strength 
varies  directly  as  the  width. 

54.  There  are  two  beams  of  equal  length  and  depth ;  one 
9  inches  in  width,  the  other  7\  inches ;  the  wider  beam  will 
sustain  47  cwt. ;  what  weight  will  the  narrower  beam  sustain  ? 


STIFFNESS    OF    BEAMS    TO    RESIST   FLEXURE.  203 

55.  There  are  two  beams  of  equal  depth ;  one  measures  20 
feet  in  length,  and  11  inches  in  width,  and  will  sustain  94 
cwt. ;  the  other  beam  is  14  feet  in  length  and  10  inches  in 
width  ;  Avhat  weight  will  it  sustain  ? 

56.  There  are  two. beams  of  the  same  width,  one  measures 
16  feet  in  length,  and  10  inches  in  depth,  and  Avill  sustain  66 
cwt. ;  the  other  is  18  feet  long,  and  12  inches  in  depth;  what 
weight  will  it  sustain  ? 

It  is  sometimes  desirable  to  know  how  the  strength  of  a 
beam  will  vary  by  removing  the  point  on  which  the  pressure 
is  made,  as  in  the  following  example. 

57.  A  beam  20  feet  in  length  will  sustain,  at  its  centre,  a 
weight  of  44  cwt. ;'  what  weight  will  it  sustain  applied  7  feet 
from  one  end  ? 

The  following  formula  will  give  the  variation  in  the 
strength. 

As  the  product  of  the  two  unequal  sections  of  the  beam, 
(in  this  case  13X7,)  is  to  the  square  of  half  the  length,  so  is 
the  weight  which  the  beam  will  sustain  at  the  centre,  to  the 
w^eight  it  will  sustain  at  the  other  given  point. 

58.  A  beam  24  feet  in^length,  will  sustain  at  its  centre  56 
cwt. ;  what  weight  will  it  sustain  at  the  dist^ce  of  9  feet 
from  one  end  ? 

59.  A  beam  28  feet  in  length  will  sustain  at  its  centre  33 
cwt. ;  what  weight  will  a  beam  of  the  same  width  and  length, 
and  of  f  the  depth  of  the  former,  sustain  at  the  distance  of 
10  feet  from  the  end  ? 

Stiffness  of  Beams  to  Resist  Flexure. 

The  stiffness  of  beams  of  the  same  length  and  width  varies, 
as  the  cuhe  of  the  depth.  If  the  depth  is  the  same,  the  stiff-' 
ness  varies  as  the  width. 

60.  There  are  two  beams  of  equal  length  and  width,  one 
is  8  inches  in  depth,  the  other  11  inches  ;  if  it  require  30 
cwt.  to  bend  the  former  one  inch,  what  weight  will  it  require 
to  bend  the  latter  one  inch  ? 

61.  There  is  a  stick  of  timber  8  inches  by  6;  if  it  require 
24  cwt.  to  bend  it  2  inches  when  lying  flat,  what  weight  will 
bend  it  2  inches  when  turned  up  on  the  edge  ? 


204  BUSINESS    FORMS    AND    INSTRUMENTS. 

G2.  If  10  cwt.  will  bend  the  stick  just  described  1|-  inches 
when  it  lies  flat,  what  weight  will  be  requisite  to  bend  it  1^ 
inches,  when  turned  up  on  the  edge  ? 

63.  There  is  a  board  12  inches  wide,  and  1  inch  in  thick- 
ness ;  what  is  the  ratio  of  its  strength  when  lying  flat,  sup- 
ported at  the  ends,  to  its  strength  when  turned  edgewise  ? 

64.  If  it  require  12  lbs.  to  bend  the  same  board  ^  an  inch, 
when  lying  flat,  how  much  will  it  require  to  bend  it  ^  an  inch 
when  turned  edgewise  ? 


SECTION    XLV. 

BUSINESS  FORMS  AND  INSTRUMENTS. 

Promissory  Notes. 
1.   On  Demand,  with  Interest. 

$500.  —  Boston,  March  1,  1846.  For  value  received,  I 
promise  A.  B.,  to  pay  him,  or  his  order,  five  hundred  dol- 
lars, on  demand,  with  interest.  T.  M. 

2.   On  Time,  with  Interest, 

$200.  —  Boston,  March  1,  1846.  For  value  received,  I 
promise  A.  B.,  to  pay  him,  or  his  order,  two  hundred  dol- 
lars, in  three  months,  with  interest.  T.  M. 

3.   On  Time,  without  Interest. 

$400. — Boston,  March  1,  1846.  For  value  received, 
I  promise  A.  B.,  to  pay  him,  or  his  order,  four  hundred  dol- 
lars, in  sixty  days  from  date.  I.  M. 

4.  Payable  by  Instalments,  with  Periodical  Interest. 

$1000.  —  Boston,  March  1,  1846.  For  value  received,  I 
promise  A.  B.  to  pay  him,  or  his  order,  one  thousand  dollars, 
as  follows,  viz. ;  —  two  hundred  dollars  in  one  year ;  two  hun- 
dred dollars  in  two  years ;  and  six  hundred  dollars  in  three 
years,  from  this  date,  with  interest  semi-annually.     I.  M. 


BUSINESS    FORMS   AND    INSTRUMENTS.  205 

Remarks  on  Promissory  Notes. 

Tyiien  the  words  "  or  order,"  are  inserted  in  a  note,  the 
holder  of  the  note  may  endorse  it,  that  is,  write  his  name  on 
the  back  of  it,  and  pass  it  to  a  third  person,  who  can  collect 
it  in  the  same  manner  as  if  he  were  the  original  holder.  If 
the  maker  of  the  note  neglects  to  pay,  the  holder  may  collect 
it  of  the  endorser. 

If  the  words  "or  bearer,"  are  inserted  instead  of  "or 
order,"  any  person  who  has  possession  of  the  note  may  collect 
It  of  the  maker.  Such  a  note  would  be  like  a  bank  note, 
which  passes  from  hand  to  hand,  without  endorsement. 

A  note,  in  order  to  be  legal  in  the  first  holder's  hands,  must 
be  for  value  received.  A  note,  therefore,  given  to  pay  a  debt 
incurred  in  gambling  or  betting,  cannot  be  collected  by  law, 
unless  it  has  passed  into  the  hands  of  an  innocent  holder. 

When  a  note  contains  the  promise  to  pay  interest  annually, 
and  the  interest  is  not  collected  annually,  the  law  does  not 
permit  the  holder  to  draw  compound  interest.  The  holder 
may  compel  the  payment  of  the  interest  when  it'  becomes 
due,  but  if  he  neglect  to  do  this,  he  can  recover  only  Simple 
Interest. 

When  a  note  is  given  to  pay  in  a  certain  commodity,  as 
wood,  grain,  &c.,  if  the  note  is  not  paid  when  due,  the  holder 
may  compel  the  payment  of  the  equitable  value  of  the  com- 
modity in  money.  The  reason  of  this  is,  that  it  is  supposed 
that  the  commodity  may  have  a  value  to  the  holder  at  the 
time  when  it  is  promised,  which  it  will  lose,  if  not  paid  then. 

Receipts.  - 

1. — A  general  Form. 

$500.  —  Boston,  March  1,  1846.  Received  of  O.  P.  the 
sum  of  five  hundred  dollars,  in  full  of  all  demands  against 
him.  A.  B. 

2.  —  For  Money  paid  hy  another  Person. 

$300.  — Boston,  March  1,  1846.  Received  of  O.  P.,  by 
the  hand  of  Y.  Z.,  three  hundred  dollars,  in  full  payment  for 
a  chaise  by  me  sold  and  delivered  to  the  said  O.  P. 

A.  B. 
18 


206  BUSINESS    FORMS    AND    INSTRUMENTS. 

3.  —  For  Money  received  for  Another, 

$700.  —  Boston,  March  1, 1846.  Received  of  O.  P.  seven 
hundred  dollars,  it  being  for  the  balance  of  account  due  from 
said'O.  P.  to  Y.  Z.  A.  B. 

4.  —  In  Part  of  a  Bond. 

$3000.  —  Boston,  March  1,  1846.  Received  of  O.  P.  the 
sum  of  three  thousand  dollars,  being  a  part  of  the  sum 
of  five  thousand  dollars  due  from  said  0.  P.  to  me  on  the 
day  of .  A.  B. 

5.  —  For  Interest  due  on  a  Bond, 

$600.  —  Boston,  March  1,  1846.  Received  of  O.  P. 
six  hundred  dollars,  due  this  day  from  him  to  me  as  the  an- 
nual interest  on  a  bond,  given  by  said  O.  P.  to  me  on  the  1st 
of  May,  1831,  for  the  payment  to  me  of  ten  thousand  dollars 
in  three  years,  with  interest  annually.  A.  B. 

6.  —  On  Account. 

$50.  —  Boston,  March  1,  1846.  Received  of  O.  P.  fifty 
dollars,  for  which  I  promise  to  account  to  him  on  a  settlement 
between  us.  ,  A.  B. 

7.  —  Of  Papers, 

Boston,  March  1,  1846.  Received  of  O.  P.  several 
contracts  and  papers,  which  are  described  as  follows  ;  —  [^de- 
scribe the  papers  ;]  which  I  promise  ta  return  to  the  said  O. 
P.  on  demand.  ,  A.  B. 

Order  at  Sight. 

Boston,  April  18,  1846.  At  sight,  pay  to  the  order  of 
John  Brown,  one  thousand  dollars,  value  received,  which 
place  to  account  of  your  obedient  servants,      A.  W.  &  Ck). 

Jacob  Smith,  Esq.,  New  York. 

Order  on  Time. 

Boston,  April  18,  1846.  Six  months  after  date,  pay  to 
the  order  of  John  iBrown,  one  thousand  dollars,  value  re- 


BUSINESS    FORMS    AND    INSTRUMENTS.  207 

ceived,  which  place  to  the   account  of  your  obedient   ser- 
vants, A.  W.  &  Co. 
Jacob  Smith,  Esq.,  New  York. 

Eemarks.  If  J  B.  present  this  order  to'' J.  S.,  and  J.  S.  write  his  name 
across  the  face  of  it,  it  J)ecomes  what  is  called  an  acceptance.  J.  S. 
'agrees  to  pay  it  at  the  date  named. 

If  J.  B.  writes  his  name  on  the  back  of  the  acceptance,  it  becomes  ne- 
gotiable ;  he  may  pass  it  to  a  third  person,  who  may  endorse  it,  and  pass 
it  to  a  fourth.  All  those  whose  signatures  are  on  the  order  are  bound  for 
its  payment ;  the  acceptor  to  the  drawer ;  the  acceptor  and  di-awer  to  the 
first  endorsor;  and  they  and  each  endorser  to  the  one  succeeding  him, 
and  the  last  endorser,  and  all  previous  parties,  to  the  holder. 

Award  by  Referees. 

We,  the  undersigned,  appointed  by  agreement  of  the  par- 
ties herein  named,  having  met  the  parties,  and  heard  their 
several  allegations,  arguments  and  proofs,  and  duly  considered 
the  same,  do  award  and  determine  that  A.  B.  shall  recover 

of  C.  D.  the  sum  of together  with  all  the  costs  of  this 

reference,  which  are  to  the  amount  of ;  and  that  this 

shall  be  final  and  in  full  of  all  claims  and  dues  of  the  parties 
on  matters  herein  referred  to  us.  I.  M. 

R.  N. 

L.  S. 

Letter  op  Credit  for  (xOODs. 

Boston,  March  1,  1846. 
Messrs.  Y.  &  Z.,  Merchants,  Baltimore. 

Gentlemen,  —  Please  to  deliver  Mr.  C.  D.,  of ,  or 

to  his  order,  goods  and  merchandize  to  an  amount  not  exceed- 
ing in  value  in  the  whole,  one  hundred  dollars  ;  and  on  your 
so  doing,  I  hereby  hold  myself  accountable  to  you  for  the  pay- 
ment of  the  same,  in  case  Mr.  C.  D.  should  not  be  able  so  to 
do,  or  should  make  default,  of  which  default  you  are  required 
to  give  me  reasonable  and  proper  notice. 

Your  obedient  servant,  A.  B. 

A  letter  of  credit  for  money  may  be  given  in  the  general  form  of  the 
above ;  specifying,  in  the  letter,  the  amount  of  credit  granted. 

Power  op  Attorney. 
Know  all  men  by  these  presents,  that  I,  A.  B.,  of 


do  hereby  appoint  C.  D.,  of ,  to  be  my  sufficient  and 

lawful  attorney,  to  act  for  me,  and  in  my  name,  [here  state 


208       THE    STANDARD    OF   WEIGHTS   AND    MEASURES. 

the  objects  for  which  he  is  to  act.]  And  for  the  purposes 
aforesaid,  I  hereby  grant  unto  my  said  attorney  full  power 
to  execute  all  needed  legal  instruments,  to  institute  and  pros- 
ecute all  claims  in  my  behalf,  to  defend  all  suits  against  me, 
to  submit  to  arbitration,  or  settle  all  matters  in  dispute,  and 
to  do  all  such  acts  as  he  shall  think  expedient  for  the  full  ac- 
complishment of  the  objects  for  which  he  is  appointed  my 
attorney,  as  fully  as  I  might  myself  do  them  if  present ;  and 
all  acts  done  by  the  said  C.  D.~my  attorney,  under  authority 
of  this  appointment,  I  will  ratify  and  confirm. 

In  testimony  whereof,  I  hereby  set  my  hand  and  seal,  this 
'  day  of ,  in  the  year  — 


Signed,  sealed,  and  delivered, 
in  the  presence  of 

a  N. 

W.  F. 


A.  B.     [l.  s.] 


SECTION    XLVI. 

ON  THE  STANDARD  OF  WEIGHTS  AND  MEASURES. 

In  the  earlier  states  of  society,  tl  e  standard  of  weights  and 
measures  was,  of  necessity,  very  indefinite  and  fluctuating. 
In  one  nation,  it  was  one  thing,  —  in  another  nation,  another; 
and  in  no  case  was  it  deserving  of  a  very  high  degree  of  con- 
fidence. 

Sometimes  the  length  of  the  king's  foot  was  the  standard 
for  all  measures  of  length ;  again,  the  length  of  the  king's 
arm  from  the  elbow  to  the  extremity  of  the  fingers,  was  made 
the  standard. 

The  length  t)f  journeys  was  measured  by  the  hours  or 
days  employed  in  performing  them,  or  by  the  number  of 
steps  taken. 

In  land  measure  the  standard  was,  what  a  yoke  of  oxen 
could  plough  in  a  day,  when,  in  fact,  one  yoke  might  plough 
twice  as  much  as  another. 

In  dry  measure,  it  was  as  much  as  a  man  could  conve- 
niently carry,  without  first  deciding  how  strong  the  man 
should  be. 


THE    STANDARD    OF    WEIGHTS    AND    MEASURES.        209 

In  weight  the  standard  was,  what  a  man  could  hold  and 
swing  in  his  hand. 

Sometimes  vegetables  were  taken  as  measures,  as  "  thi-ee 
barley  corns  make  one  inch."  But  barley  corns  do  not  all 
grow  of  exactly  equa.l  length,  any  more  than  the  feet  and 
arms  of  kings. 

As  science  advanced,  and  commerce  became  farther  ex- 
tended among  diiferent  nations,  the  mischiefs  of  these  vague 
and  fluctuating  methods  of  measurement  became  more  and 
more  deeply  felt. 

But  it  was  far  easier  to  see  the  faults  of  the  old  system, 
than  to  devise  a  new  one  that  should  be  perfect.  "What  ob- 
ject could  be  selected  as  an  ultimate  standard  for  all  weights 
and  measures  ? 

We  have  seen  that  the  parts  of  animals  or  of  vegetables, 
are  too  liable .  to  change  to  deserve  any  confidence.  If  some 
arbitrary  standard  should  be  adopted,  as  a  foot,  or  yard,  and 
this  measure  should  be  kept  as  the  standard,  by  which  all 
others  should  be  tried,  what  security  could  there  be  that  it 
would  never  be  altered  by  fraud,  or  destroyed  by  accident  ? 
Or,  if  some  natural  distance  were  taken,  as  the  distance  be- 
tween two  points  of  some  well  known  rock  or  cliff,  this  dis- 
tance might  vary  with  a  change  of  temperature,  or  be  altered 
by  some  convulsion  of  nature. 

We  will  proceed  to  give  a  short  account  of  the  English 
system  of  weights  and  measures,  adopted  by  their  Act  of 
Uniformity/,  which  took  effect  Jan.  1,  1826.  To  begin  with 
measures  of  capacity ;  all  English  measures  of  capacity, 
whether  for  liquors  or  grain,  are  referred  to  the  standard  im- 
perial gallon.  This  gallon  contains  217^  cubic  inches.  From 
this  gallon,  quarts,  pints,  and  gills  are  obtained  by  subdivi- 
sion ;  and  pecks  and  bushels  by  multiplication.  Hence,  you 
can  find  the  number  of  cubic  inches  in  an  English  quart,  pint, 
peck,  or  bushel.  Thus  the  adoption  of  the  imperial  gallon 
introduces  entire  uniformity  into  all  English  measures  of 
capacity.  It  refers  them  all  ultimately  to  the  cubic  inch. 
We  must  now  inquire,  what  has  been  done  to  fix  the  measure 
of  the  inch  ?  for,  if  there  is  any  error  or  variation  here,  it 
will  render  false  all  the  measures  of  capacity  which  depend 
upon  it. 

To  determine  the  measure  of  the  inch,  it  is  made  by  law 
g^  of  the  standard  yard.  That  standard  yard  is  a  straight 
18*      _ 


210       THE    STANDARD    OF    WEIGHTS    AND    MEASURES. 

brass  rod  in  the  custody  of  the  Clerk  of  the  House  of  Com- 
mons. The  yard  is  the  distance  on  that  rod  between  the 
centres  of  the  points  in  the  two  gold  studs  or  pins  in  the  rod. 
And  as  heat  would  make  the  yard  longer,  and  cold  would 
make  it  shorter,  the  law  requires  that  it  shall  be  used  when 
it  is  of  the  temperature  of  62°  (Fahrenheit.)  This  standard 
yard,  however,  may  be  destroyed  by  accident.  We  must 
then  inquire  for  a  still  more  permanent  standard.  To  effect 
this,  the  law  declares  that  the  standard  yard,  if  destroyed, 
may  be  restored,  by  making  it  f ff §§§  of  the  length  of  a  pen- 
dulum, that  vibrates  seconds  in  the  latitude  of  London,  in  a 
vacuum,  at  the  level  of  the  sea.  If  all  these  conditions  are 
fulfilled,  a  pendulum  that  vibrates  seconds  must  have  an  ab- 
solutely invariable  length. 

Thus  we  have  brought  the  whole  system  of  measures  back 
to  seconds,  as  the  standard.  The  whole  scheme  now  depends 
upon  seconds  being  of  an  invariable  length. 

Seconds  are  parts  of  a  year ;  the  year  is  not  made  up  by 
multiplying  seconds,  but  seconds  are  obtained  by  dividing  the 
year.  If,  then,  the  year  is  of  a  fixed  length,  seconds  are  so. 
Now  the  year  is  the  time  of  the  revolution  of  the  earth  round 
the  sun.  It  is  the  same,  without  change,  from  one  year  to 
another,  and  from  century  to  century. 

Thus  the  whole  system  of  measures  has  been  brought,  for 
its  ultimate  standard,  to  the  unalterable  period  of  the  earth's 
revolution  round  the  sun.    — 

We  will  now  retrace  the  steps  of  this  investigation,  be- 
ginning with  the  primary  standard,  the  earth's  yearly  revo- 
lution. 

The  time  of  the  earth's  revolution  round  the  sun  is  always 
the  same ;  therefore,  a  second,  which  is  a  certain  part  of  this 
time,  is  an  exact  measure.  If  the  second  is  a  fixed  measure, 
then  the  pendulum  which,  under  the  same  circumstances, 
vibrates  seconds,  is  of  a  fixed  length.  If  the  length  of  the 
pendulum  vibrating  seconds  is  fixed,  the  length  of  the  stand- 
ard yard  is  fixed,  for  it  is  ff ^§§f  of  the  pendulum.  If  the 
standard  yard  is  fixed,  the  inch  is  fixed,  consequently  the 
cubic  inch,  the  gallon,  quart,  pint,  gill,  and  bushel. 

In  the  preceding  investigation  no  mention  has  been  made 
of  the  standard  of  weight.  It  is  obtained  by  making  a  cubic 
inch  of  distilled  water  equal  to  252.458  grs.,  of  which  5760 
make  a  pound  troy,  and  7000  make  a  pound  avoirdupois. 


THE   STANDARD    OP   WEIGHTS   AND    MEASURES.       211 

Thus  weights  and  measures  are  alike  brought  to  an  unal- 
terable standard. 

The  imperial  gallon  contains 277^  cubic  inches. 

The  Winchester*  gal.,  wine  measure,  •  •  •  -231  " 

"  "  «    •  beer  measure, 282  " 

The  imperial  gallon  of  water  weighs  10  lbs.  avoirdupois. 

The  system  of  weights  and  measures  established  by  law 
in  the  United  States,  is  very  nearly  the  same  as  the  English. 
The  gallon,  United  States  measure,  contains  9  lbs.  14  oz.  of 
water.  This  is  the  legal  standard  for  all  measures,  dry  and 
liquid.  In  many  parts  of  the  country,  however,  especially  in 
the  interior,  the  legal  -standard  has  not  supplanted  the  sys- 
tem derived  in  earlier  times  from  the  English. 

In  France,  where  the  system  of  weights  has  been  carried 
to  greater  perfection  than  in  any  other  country,  the  decimal 
ratio  is  adopted  in  all  denominations.  In  some  cases,  how- 
ever, there  is  still  retained  some  part  of  the  old  system, 
combined  with  the  decimal. 

In  obtaining  an  ultimate  standard  of  measure,  the  French 
measured  one  quarter  of  a  meridian  line  of  longitude. 

One  ten  millionth  part  of  this  arc  they  made  the  basis  of 
their  system  of  measures.  This  standard,  the  metre,  is  3.28 
feet.  The  lower  denominations  are  made  by  successive 
divisions  of  this,  by  10,  100,  &c. ;  and  the  higher  by  multi- 
plication. 

The  following  table  presents  the  French  decimal  weights 
and  measures,  with  the  English  equivalents. 

French  Long  Measure, 

feet. 

10  mellimetres    make  •  •  1  centimetre,    .0328 

10  centimetres 1  decimetre, .328 

10  decimetres  • 1  metre, 3.28 

10  metres 1  decametre, 32.8 

10  decametres 1  hectometre, 328 

10  hectometres 1  kilometre, 3280 

10  kilometres 1  myreametre 32800 


*  Winchester,  so  called  because  the  standard  measures  were  kept  at 
Winchester. 


212        THE    STANDARD    OF    WEIGHTS    AND    MEASURES. 


French  Square  Measure. 

The  unit  square  measure  is  the  are,  which  is  the  square  of 
the  decametre  ;  consequently,  it  is  the  square  of  32.8  feet,  — a 
little  less  than  4  square  rods. 

This  ^unit  is  multiplied  for  the  'higher  denominations,  and 
divided  for  the  lower,  in  the  same  way  as  the  metre. 


French  Decimal   Weight. 


10  milligrammes  make  1 

10  centigrammes 1 

10  decigrammes    1 

10  grammes 1 

10  decagrammes 1 

10  hectogrammes    •  •  •  •  1 

10  kilogrammes 1 

10  myriagrammes  •  •  •  •  1 
10  quintals  •  • 1 


centigramme,  •  • 
decigramme,   •  • 

gramme, 

decagramme,  •  •  < 
hectogramme,  • 
kilogramme,  •  • 
myriagramme,  • 

quintal,     

million, 


gre.  Troy. 

.1543402 
1.543402 
15.43402 
154.3402 
1543.402 
15434.02 
154340.2 
1543402. 
5434020. 


APPENDIX. 


The  examples  that  follow  are  miscellaneous,  and  designed  to  carry  still  farther 
the  practice  in  Written  Arithmetic. 

1.  James  Ball  bought  of  Amos  SeAvall  three  pieces  broadcloth,  measur- 
ing 12^,  13,  and  24  yards,  at  $4.87|-  per  yard;  five  pieces  kerseymere, 
measuring  24|-,.25,  27,  265^,  and  26  yards,  at  67  cts.per  yard ;  eight  pieces 
cotton  sheeting,  measuring  33  yards  each,  at  9\  cents  per  yard,  ^  per  cent 
off.     What  was  the  amount  of  the  bill  1 

2.  Bought  of  John  Jones,  on  six  months'  credit,  47  yards  broadcloth, 
at  f4.3l  per  yard ;  16  yards  vestings,  at  $1.15  per  yard;  63^  yards  satinet, 
at  62^  cents  per  yard ;  5^  pieces  sheeting,  containing  33  yards  a  piece,  at 
8|  cents  per  yard.  John  Jones  agrees,  if  paid  in  cash,  to  deduct  4  per 
cent,  from  the  bill ;  what  is  the  cash  amount  of  the  bill  1 

3.  Bought  of  Asa  Wood,  on  six  months'  credit,  45  ban-els  flour,  at  $5.37 
per  barrel;  four  hhds.  molasses,  containing  124^,  131,  134,  and  136  gal- 
lons, at  27^  cents  per  gallon ;  five  bags  coffee,  containing  541^,  56,  49^,  62, 
and  65j  lbs.  at  8;^  cents  per  lb.  Gave,  in  payment  of  the  above,  a  note 
payable  in  six  months ;  what  was  the  cash  value  of  the  note  when  it  was 
given,  reckoning  interest  at  6  per  cent.  1 

4.  A  bought  of  B,  on  six  months'  credit,  goods  with  the  amount  aa 
follows, 

April  3,  1845. $254.75 

June  8,      " 135.00 

Aug.  1,       " 200.00 

Sept.  14,    " 168.25 

At  what  date  shall  A  makcan  equated  payment  of  the  whole  amount  ? 

5.  A  gives  B  a  note  for  $600,  payable  in  six  months ;  what  is  the  cash 
value  of  the  note,  two  months  after  date,  reckoning  the  interest  at  6  per 
cent.  1 

6.  A  man  agrees  to  dig  and  stone  a  well,  on  the  following  terms ;  $1.00 
per  foot  for  the  first  ten  feet,  $2.50  per  foot  for  the  second  10  feet,  and 
$4.00  per  foot  for  the  remainder,  till  he  finds  a  supply  of  water ;  for  every 
foot  of  rock  through  which  he  digs  he  shall  receive  double  pay.     He  digs 

.  42  feet  in  all,  and  through  rock  from  17  to  31|-  feet  from  the  surface ;  what 
pay  is  he  entitled  to  for  the  whole  ? 

7.  A  man  engages  to  build  160  rods  of  road,  one  half  for  $1.42  per 
rod,  the  other  half  for  $1.83  per  rod;  he  hires  93  days  of  men's  labor  at 
84  cents  per  day,  pays  for  board  of  the  same  at  $1.50  per  week  of  six 
days;  pays  for  tools  and  repairs  $11.60;  he  works  himself,  with  4  oxen 


214 


APPENDIX. 


and  his  son,  34  days.  "What  wages  will  he  receive  per  day  for  himself, 
for  his  oxen,  and  for  his  son,  allowing  for  the  4  oxen  as  much  as  for  him- 
self, and  for  his  son  half  as  much  1 

8.  A  bought  a  lot  of  standing  wood  for  105  dollars,  and  agreed  with 
B  to  cut  and  haul  it  to  market  for  three-fifths  of  the  proceeds;  there 
were  54  cords  of  pine  which  was  sold  for  $2.84  per  cord,  and  61  cords  of 
hard  wood  which  sold  for  $4.75  per  cord.  Did  A  gain  or  lose,  and  how 
much  ?  B  labored  himself  35  days,  employed  one  yoke  of  oxen  24  days, 
and  hired  sixty-eight  days  of  men's  labor  at  85  cents  per  day  5  what  pay 
does  he  receive  per  day  for  himself  and  for  his  oxen,  allowing  for  his 
oxen,  per  day,  two-thirds  as  much  as  for  himself  *? 

9.  A  and  B  bought  a  quantity  of  grass,  ready  for  cutting,  for  26  dol- 
lar, for  which  they  paid  13  dollars  apiece.  In  cutting  and  curing  it  A 
fiirnished  3  days  of  hired  men's  labor  and*  worked  himself  2^  days,  B 
hired  7  days  of  men's  labor,  a  team  1^  days,  and  worked  himself  2^  days. 
There  were  62-  tons  of  hay ;  what  is  each  one's  share  of  the  hay,  allowing 
for  the  labor  $1.00  per  day,  and  for  the  whole  work  of  the  team  $1.50  ? 

10.  A  agrees  to  dig  and  stone  a  cellar,  7  feet  deep;  it  is  to  be  13  feet' 
wide  and  16  feet  long  inside  the  walls,  which  are  to  be  2  feet  in  thickness. 
He  is  to  receive  for  digging  22  cents  a  cubic  yard  for  earth,  and  $1.55  a 
cubic  yard  for  rock,  and  for  stoning  87  cents  for  every  perch  of  25  cubic 
feet,  for  which  the  stone  is  found  in  digging  the  cellar,  and  $1.50  a  perch 
when  he^  has  to  bring  the  stone  from  another  place.  In  digging  the  cel- 
lar he  digs  7^  cubic  yards  of  rock,  which  furnishes  stone  for  7^  cubic 
yards  of  wall ;  what  is  he  to  receive  for  the  whole  job  '? 

11.  A  man  agrees  to  dig  a  canal  10  rods  long,  6  feet  in  depth,  33  feet 
wide  at  the  top,  and  24  feet  wide  at  the  .bottom,  for  10  cents  per  cubic 
yard ;  what  sum  will  the  work  amount  to  ? 

12.  What  is  the  value  of  six  loads  of  wood  at  $4.67  per  cord,  measur- 
ing as  follows  ;  1st,  8  ft.,  4  ft.  3  in. ;  3  ft..  10  in. ;  2d,  8  ft.  4  in.,  4  ft.  1  in., 
4  ft. ;  3d,  8  ft.  2  in.,  4  ft.  7  in.,  3  ft.  11  in. ;  4th.  8  ft,  4  ft.  2  in.,  4  ft;  5th, 
8  ft.  2  in.,  3  ft.  10  in.,  3  ft.  9  in. ;   6th,  8  ft.  6  in.,  4  ft.,  4  ft.  4  in.  1 

13.  What  will  be  the  dimension,  at  the  two  ends,  of  the  largest  square 
stick  of  timber  that  can  be  hewn  from  a  round  log  3  feet  in  diameter  at 
the  larger  end,  and  2  feet  in  diameter  at  the  smaller  ? 

14.  What  will  be  the  solid  contents  of  such  a  stick,  if  it  is  16  feet  in 
length  ? 

15.  What  will  be  the  width  at  the  two  ends  of  the  largest  stick  of  tim- 
ber that  can  be  hewn  from  a  log  3  feet  in  diameter  at  the  larger  end,  and 
2  feet  in  diameter  at  the  smaller,,  if  the  stick  is  hewn  10  inches  in  thick- 
ness through  its  whole  length  1 

16.  What  will  be  the  solid  contents  of  such  a  stick,  if  it  is  20  feet  in 
length  1 

17.  There  are  three  pieces  of  cloth;  the  length  of  the  first  is  to  that  of 
the  second  as  3  to  2,  the  length  of  the  second  is  to  that  of  the  third  as  4 
to  15,  and  the  length  of  the  three  added  together  is  50  yards  ;  what  is  the 
length  of  each  piece  "? 

18.  A  man  bought  a  chaise  and  paid  20  dollars  for  repairing  it,  he  then 
Bold  it  for  one  fifth  more  than  he  gave,  and  found  that,  allowing  one  dol- 
lar for  his  own  trouble  in  the  business,  he  had  lost  thirteen  dollars ;  what 
did  he  give  for  the  chaise  ? 

19.  A  gentleman  began  his  preparation  for  college  at  a  certain  age, 
and  spent  in  school  and  at  college  half  as  many  years  as  he  had  lived 


APPENDIX.  215 

before ;  he  then  went  to  Europe,  and  after  spending  there  one  ninth  as 
many  years  as  his  age  amounted  to  when  he  left  Europe,  spent  in  his  pro- 
fession one  thu-d  as  many  years  as  he  had  lived  when  he  entered  it,  and 
was  then  36  years  old ;  at  what  age  did  he  begin  his  preparation  for 
college  ? 

20.  One  half  Of  three  fourths  of  A's  age  equals  one  sixth  of  B's  age, 
and  the  sum  of  their  ages* is  78 ;  what  is  the  age  of  each  ? 

21.  Three  fourths  of  the  liquor  in  a  cask  equals  five  sixths  of  what  has 
leaked  out,  and  the  whole  before  any  leaked  out  was  sixty  gallons ;  how 
much  is  there  in  the  cask  7 

22.  Reduce  5  pence  to  the  decimal  of  a  shilling,  carrying  the  decimal 
to  the  sixth  figure. 

23.  Reduce  9^  gallons  to  the  decimal  of  a  barrel,  wine  measure,  carry- 
ing the  decimal  to  the  tenth  figure. 

24.  Reduce  7^  quarts  to  the  decimal  of  a  bushel. 

25.  What  is  the  least  common  multiple  of  784,  1386,  and  12351 

26.  What  are  the  prime  numbers  between  1010  and  1020 1 

27.  What  are  the  prime  factors  of  2326  ? 

28.  Reduce  — r  — -    and  — -  to  simple  fractions,  with  a  common 

.  22^   19  30i  ^ 

denominator. 

29.  What  is  the  value  in  shillings  and  pence  of  the  following  decimals 
when  added  together,  £.0431,  .67142s.,  .73462c?.? 

30.  What  is  the  interest  of  $642.25  for  1  year  and  3  months  at  7^ 
per  cent.  1 

31.  What  is  the  interest  of  $954.30  for  2  years,  4  months,  and  21  days, 
at  5  per  cent.  1 

32.  What  is  the  present  worth  of  a  note  of  $640.50  due  in  3  months  1 

33.  What  is  the  present  worth  of  a  note  of  $1263.00  due  in  3^  months 
at  5  per  cent,  interest  ? 

34.  Boston,  June  14,  1836.  Eor  value  received  I  promise  to  pay 
John  Ball,  or  order,  three  hundred  and  sixty^five  dollars  in  four  years 
with  interest  annually.  James  Fkost. 

If  no  interest  is  paid,  and  the  note  is  renewed  annually,  what  will  be 
the  amount  of  the  note  four  years  after  date  ? 

35.  New  York,  Jiily  1,  1840.  For  value  received  I  promise  to  pay 
Abel  Jones  or  order  five  hundred  and  forty  dollars,  on  demand,  with  in- 
terest. .  John  Fkost. 

Endorsements, 

Oct.  3,  1840, $63  00 

Feb.  4, 1841, 120.00 

June  1,  1841, 60.00 

Sept.  15, 1841, 200.00 

What  will  be  due  Feb.  14,  1842,  at  seven  per  cent,  interest? 

36.  A  owes  B  $400.00,  due  in  2  months ;  $320.00,  due  in  3  months ; 
$600.00,  due  in  4^  months ;  what  is  the  equated  time  for  the  payment  of 
the  whole  1 

37.  What  is  the  present  worth  of-  a  bank  note  for  800  dollars  payable 
in  three  months  ? 

38.  What  is  the  present  worth  of  a  bank  note  for  346  dollars  payable 
in  six  months  ? 

39.  For  what  sum  must  I  give  a  note  to  a  bank  payable  in  three  months 
in  order  to  obtain  674  dollars  ? 


216  APPENDIX. 

40.  Bought  seven  100  dollar  shares  of  Bank  stock  at  6^  per  cent,  ad- 
■  vance,  gave  in  payment  nine  60  dollar  shares  of  Rail  Road  Stock  at  3 

per  cent,  discount,  and  a  bank  note  payable  in  60  days  ;  what  was  the  faco 
of  the  note  ? 

41.  How  many  square  inches  are  there  in  15|-  square  rods  ? 

42.  What  is  the  cost  of  plastering  the  sides,  ends,  and  ceiling  of  a  room 
22^  feet  long,  17|  feet  wide,  and  11  feet  1  inch  in  height,  at  18  cents  per 
square  yard,  making  no  deduction  for  windows,  doors,  or  wood  work  ? 

43.  There  is  a  house  40  feet  in  length,  and  26  feet  in  breadth ;  from 
the  beam  to  the  ridgepole  is  1 1  feet ;  the  roof  projects  7  inches  beyond 
the  walls  at  each  end,  and  the  line  of  the  eaves  is  8  inches,  measured 
horizontally,  from  the  side  walls ;  how  many  square  feet  are  there  in  the 
roof? 

44.  A  painter  agrees  to  paint  tlie  outside  of  a  house  which  is  44  feet 
long,  28  feet  wide,  22^  feet  in  height  to  the  top  of  the  beam,  and  11  feet 
8  inches  from  the  beam  to  the  ridgepole,  for  44  cents  per  square  yard ; 
what  is  he  entitled  to  for  the  job,  making  no  deduction  for  windows,  and 
no  addition  for  cornices  or  other  projections  1 

45.  What  is  the  square  root  of  9743  to  three  places  of  decimals? 

46.  What  is  the  square  root  of  17431  to  two  places  of  decimals  ? 

47.  The  base  of  a  right  angled  triangle  is  744  feet,  the  hypotenuse  834 
feet ;  what  is  the  perpendicular,  to  two  places  of  decimals  ? 

48.  The  base  of  a  ti-iangle  is  76  rods,  the  sum  of  the  hypotenuse  and 
perpendicular  is  186  rods ;  what  is  the  length  of  the  hypotenuse  ? 

49.  From  a  cylinder,  twelve  inches  in  diameter,  it  is  desired  to  cut  the 
largest  possible  four  sided  prism,  whose  opposite  sides  .shall  be  parallel, 
and  whose  width  shall  be  to  its  thickness  as  2  to  1 ;  what  will  be  its  width 
and  what  its  thickness  1 

50.  What  are  the  dimensions  of  the  largest  prism,  with  parallel  sides, 
that  can  be  cut  from  a  cylinder  12  inches  in  diameter,  making  the  width 
to  the  thickness  as  3  to  1  ? 

51.  What  are  the  dimensions  of  the  largest  prism,  with  parallel  sides, 
that  can  be  cut  from  a  sphere  15  inches  in  diameter,  making  the  length 
and  breadth  equal,  and  each  of  them  double  of  the  thickness  1 

52.  What  is  the  cube  root  of  674  to  three  places  of  decimals  i 

53.  What  is  the  cube  root  of  1736  to  two  places  of  decimals  ? 

54.  What  is  the  cube  ro'Ot  of  31  to  three  places  of  decimals  ? 

55.  Two  men  purchase  a  lot  of  land  for  750  dollars,  one  pays  $406.50, 
the  other  the  remainder;  they  expend  $341  in  equal  shares  on  its  im- 
provement, and  sell  the  land  for  $1430.00;  what  is  each  one's  sham  of 
the.  gain? 

56.  1841  are  how  many  times  four-fifths  of  76j1 

57.  How  many  bottles,  each  containing  1|  puits,  can  be  filled  from  a 
hogshead  containing  63  gallons,  allowing  a  loss  of  one  eleventh  in  the 
process  1 

58.  What  is  the  value,  in  Federal  money,  of  £456  at  9^  per  cent,  ad- 
vance 1 

59.  How  many  gallons,  each  containing  231  cubic  inches,  Avill  fill  a 
cylindrical  cistern  four  feet  in  diameter  and  five  feet  deep  ? 

60.  There  is  a  cylindrical  cistern  6  feet  deep,  eontaining  10  barrels  of 
31|^  gallons  each,  each  gallon  containing  231  cubic  inches ;  what  is  the 
diameter  of  the  cistern  1 

61 .  There  is  a  cistern,  in  the  form  of  an  inverted  cone,  eight  feet  deep, 


APPENDIX.  217 

and  of  the  fame  capacity  as  the  cistern  last  named ;  what  is  its  diameter 
at  the  top  ? 

62.  Bought  74  barrels  of  flour  at  $4.56  per  barrel,  and  after  keeping  it 
35  days  sold  it  at  $5.16  per  barrel;  what  per  cent,  did  I  gain,  allowing 
6  per  cent,  interest  on  the  money  invested  1 

63.  The  first  and  fourteenth  terms  of  an  arithmetical  series  are  3  and 
19;  what  is  the  common  increase? 

64.  The  tifth  term  of  an  arithmetical  series  is  18,  the  16th  term  is  39; 
what  is  the  first  term  ? 

65.  What  is  the  sum  of  an  arithmetical  series  the  extremes  of  which 
are  9  and  1 64,  and  the  number  of  terms  forty  1 

66.  Find  the  ninth  term  of  a  geometrical  series  whose  first  term  is  2, 
and  ratio  |. 

67.  What  is  the  fifth  term  of  a  geometrical  series  whose  second  term 
is  4,  and  ratio  §  1 

68.  James  Wildes  bought  of  John  Good, 

'  45A  bushels  Salt  at  39^  cents  per  bushel,  ^^ 

143|  lbs.  Rice  at  3|  cents  per  lb. 
43i  lbs.  Tea  at  39  cents  per  lb. 
94|  lbs.  Coffee  at  10^  cents  per  lb. 
12    bbls.  Flour  at  $5.87  per  bbl. 
What  is  the  amount  of  the  bill  1 

69.  If  he  pays  the  above  bill  by  a  bank  note,  discounted  for  60  days, 
what  must  be  the  sum  named  in  the  note  ? 

70.  Add  23 A +1814-^+ -^-f^- 

71.  Reduce  to  a  common  denominator 

2^  and  |  of  j\  of  12^  and  |  of  ^. 

72.  Bought  3  boxes  of  sugar,  containing  3  cwt.  2  qrs.  17  lbs.;  3  cwt. 
3  qrs.  1  U  lbs  :  3  cwt.  2  qrs.  22|  lbs.,  at  G^  cts.  per  lb.  Paid  in  corn  at 
57^  cts.  per  bushel ;  how  many  bushels  did  it  take? 

73.  What  is  the  solid  contents  of  a  wall  5  ft.  high;  2  ft.  3  in.  wide  at 
the  bottom,  and  1  foot  10  in.  wide  at  the  top.  and  34  ft.  in  length  1 

74.  Three  men  agreed  to  build  a  wall  6  ft.  high,  3  ft.  wide  at  the  bot- 
tom, and  2  ft.  wide  at  the  top ;  the  first  man  built  the  wall  to  the  hCight 
of  two  feet  from  the  ground,  the  second  raised  it  2  feet  more,  and  the 
third  finished  it;  what  proportional  share  of  the  pay  ought  each  to  re- 
ceive ? 

75.  There  is  a  lever,  13  ft.  in  length,  which  is  supported  by  a  fulcrum 
14  in.  from  the  end;  how  many  pounds  applied  to  the  longer  end  will 
balance  a  weiglit  of  17  cwt.  2  qrs.  at  the  shorter  end  ? 

76.  The  axle  of  a  wheel  is  13  in.  in  diameter,  the  wheel  is  11^  ft.  in 
diameter ;  what  power  applied  to  the  ch-cumfercncc  will  balance  3"^  tons 
suspended  at  the  axle  ? 

77.  If  tlic  threads  of  a  screw  are  l^-  in.  apart,  what  power  applied  at 
the  end  of  a  lever  9^  ft.  in  length  will  support  7  tons,  allowing  nothing 
for  friction  ? 

78.  With  the  same  screw,  what  power  would  support  7  tons,  making  an 
allowance  of  one  fourth  for  friction?  Reflect  whether  the  friction  in  thi.s 
case  is  in  favor  of  the  weight  or  in  favor  of  the  power. 

79.  With  the  same  screw,  what  power  would  raise  7  tons,  allowing  one 
fourth  for  friction  ?  Notice,  in  this  case,  in  which  way  the  friction  will 
operate.  19 


218 


ArPKNDlX. 


6 

1 


80.  There  are  two  right  angled  triangles  U])on  ahase  of  21  ft. in  length,- 
the  perpendicular  of  the  larger  is  9  ft.  in  length,  that  of  the  smaller  8^  ft., 
^rhat  is  the  difference  in  the  length  of  the  hypotenuse  of  the  two  triangles'? 

81.  Divide  the  number  78  in  two  such  numbers  that  the  first  shall  be 
6  more  than  one  fifth  part  of  the  second. 

v^    82.  Divide  the  number  82  into  two  such  parts  that  the  first  diminished 
)y  5  shall  equal  one  sixth  part  of  the  second. 

83.  What  is  the  value  in  Federal  money  of  £194  16s.  at  8^  per  cent, 
advance  ? 

84.  How  many  Winchester  gallons,  Avine  measure,  would  be  contaihed 
\    in  a  cubical  vat  measuring  4  ft.  each  way  ? 

\\     85.  Divide  the  number  1520  into  three  such  parts  that  twice  the  first 
shall  be  40  less  than  the  second,  and  the  second  shall  be  half  as  great  as 
the  third. 
,     86.  How  many  yards  of  lining  |  of  a  yard  wide  will  line  13|  yds.  of 

^  cloth  l^v    yds.  wide  ? 

87.  There  is  a  rectangular  field  containing  7j  acres,  the  width  of  which 
is  one  half  as  great  as  its  length ;  what  is  the  length  of  a  diagonal  line 
connecting  its  opposite  corners  ? 

88.  Around  a  rectangular  common  containing  18  acres,  the  length  of 
which  is  to  its  breadth  as  6  to  5,  a  road  runs  40  ft.  in  breadth ;  how  many 
rods  would  be  saved  in  travel  by  crossing  the  common  diagonally,  rather 
than  going  round  on  two  sides  of  it,  supposing  the  traveller  to  begin  and 
end  in  both  cases  in  the  middle  of  the  road  in  range  with  the  diagonal 
line? 

89.  What  is  the  difference  between  the  square  root  of  half  of  4^  and 
half  the  square  root  of  4,  carried  to  three  decimal  places  1 

90.  What  is  the  difference  between  the  square  root  of  one  third  of  12, 
and  one  third  the  square  root  of  1 2,  carried  to  three  places  of  decimals  1 

91.  How  many  times  |  of  171  are  equal  to  13^  times  15^  ? 

92  When  it  is  noon  in  Boston,  Lon.  71°  4'  W.  what  time  is  it  at  Liver- 
pool, Lon.  2'  59'  W.:  at  Greenwich,  Lon.  0;  at  Havre,  Lon.  0  16'  E.; 
and  at  Paris,  Lon.  2"  20'  E.  ? 

93.  Wlien  it  is  noon  at  London,  Lon.  0  5'  W.,  what  time  is  it  at  New 
York,  Lon.  74°  1'  W.;  at  Washington,  Lon.  77'  2'  W. ;  and  at  Cincin- 
nati, Lon.  84°  27'  W.  ? 

94.  A  field  in  the  form  of  an  equilateral  triangle  contains  8|  acres; 
what  is  the  length  of  one  of  its  sides  ? 

95.  Reduce  f  of  —  to  fifths.     Reduce  9  of  — -  to  fifteenths. 

11  ISi 

96.  What  is  the  amount  in  Federal  money  of  6s.  7d,,  5s.  3d.,  14s.  gd^ 
and  lis.  e^d.l 

97.  What  is  the  cube  root  of  144  to  three  places  of  decimals  ? 

98.  Wliat  is  the  weight  of  a  cylinder  of  lead  3  ft.  long  and  4  inches  in 
diameter  1 

99.  Multiply  7^  times  Hi  by  |  of  ^ 

100.  How  many  square  feet  in  a  triangle  whose  base  measures  45  feet, 
and  whose  height  is  17  fcef? 

101.  What  is  the  solid  contents  of  a  triangular  pyramid  whose  base 
measures  18  ft.  on  each  side  and  whose  height  is  23  ft. "? 

102.  What  is  the  interest  of  $546.25  for  23  mo.  13  days,  at  7^  per  ct.  ? 


APPENDIX.  219 

103.  What  number  is  that  of  which  ^  and  ^  of  it  added  together  ex- 
ceed ^  of  it  by  2  ? 

104.  What  is  the  cube  root  of  197  to  two  decimal  places  ? 

105.  What  is  the  cube  root  of  501  to  four  decimal  places  ? 

106.  What  is  tlie  43d  term  of  an  arithmetical  series  whose  first  term 
is  7^  and  common  difference  3§ "? 

107.  What  is  the  suni  of  an  arithmetical  series  of  57  terms  whose  4th 
term  is  15  and  common  difference  2^1 

108.  How  many  lbs.  of  coffee  at  11  cents  per  lb.  can  be  mixed  with 
56  lbs.  at  8  cents,  and  96  lbs.  at  9^  cents,  so  as  to  make  the  mixture  worth 
10^  cents  ? 

109.  If  a  sphere  of  gold  weigh  36  oz.  how  many  oz.  will  a  sphere  of 
silver  weigh  of  equal  size,  the  specific  gravity  being  as  given  on  p.  198  ? 

110.  What  will  be  the  weight  of  a  ball  of  iron  6  in.  in  diameter? 

111.  What  will  be  the  weight  of  the  largest  cube  that  can  be  cut  from 
a  ball  of  iron  6  in.  in  diameter  ? 

112.  What  is  the  value  in  Federal  money  of  £13  6s.,  2  guineas,  3d., 
7:|^d.,  added  together  1 

113.  How  many  times  will  a  wheel  4  ft.  3  in.  in  diameter  go  round  in 
traversing  the  circumference  of  a  circle  containing  5  acres  ? 

114.  If  a  lever  is  16  ft.  in  length,  the  weight  13  cwt.  3  qrs.  11  lbs.,  and 
the  power  94  lbs.,  what  must  be  the  distance  of  the  fulcrum  from  the 
weight  in  order  that  the  weight  and  power  may  balance  1 

115.  If  the  longer  arm  of  a  lever  be  10  ft.  and  the  shorter  arm  2  feet  in 
length,  how  must  480  pounds  be  divided  so  that  one  part  shall  be  the 
weight  and  the  other  the  power  that  will  balance  it  on  the  lever  1 

116.  Divide  f  of  J  of  16^  by  18^  times  |  of  7|. 

117.  Reduce  1  pk.  3  qts.  1  pt.  1  gill  to  the  decimal  of  a  bushel. 

118.  How  many  shillings  and  pence  in  .4562  of  &  £. 

119.  A  general  drew  up  his  army  in  a  square,  with  the  number  in  rank 
and  file  equal,  and  had  576  men  left;  he  then  increased  the  square  by 
placing  two  lines  of  men  in  front,  and  two  files  on  one  side  from  front  to 
rear,  when  he  found  he  wanted  12  men  to  complete  the  square;  how 
many  men  had  he  ? 

120.  What  number  is  that  one  third  of  which  exceeds  two  sevenths  of 
it  by  19  ? 

51  9 

121.  What  number  is  that,  —  of  which  exceeds  — •  of  it  by  31  ? 

13  23  -^ 

122.  How  many  tiles  each  8  inches  square  will  it  require  to  cover  one 
acre  1 

123.  If  the  hypotenuse  of  a  right  angled  triangle  measure  34  ft.  and 
the  base  19^,  what  is  the  measure  of  the  perpendicular? 

124.  If  the  sum  of  the  base  and  hypotenuse  is  63  ft.  and  the  perpen- 
dicular 14  ft.  how  long  is  the  base  ? 

125.  A  man  travels  South  20  miles,  then  East  15  miles,  then  South  2^ 
miles,  and  East  7  miles ;  how  far  is  he  then  from  where  he  set  out  ? 

126.  What  is  the  difference  between  the  cube  root  of  one  third  of  twelve, 
and  one  third  of  the  cube  root  of  twelve  ? 

127.  What  is  the  value  in  dollars  and  cents  of  £94  16s.  3d.-f-^3  19s. 
7d.-f-jei4  13s.  9d.? 

128.  If  a  note  of  $1000  promising  annual  interest  is  renewed  at  the 
end  of  each  year  for  five  years,  without  the  payment  of  any  interest,  what 
is  the  amount,  principal,  and  interest,  at  the  end  of  the  fifth  year  ? 


220 


APPENDIX. 


129.  What  is  the  difference  in  avoirdupois  weight  between  a  ball  of 
silver,  and  a  ball  of  gold,  each  3  in.  in  diameter? 

1.30.  There  are  three  numbers,  the  first,  plus  4,  is  equal  to  one  sixth  of 
the  second,  the  second  is  one  half  as  great  as  the  third,  and  the  sum  of 
the  three  is  186;  what  are  the  numbers? 

131.  There  are  two  numbers,  the  first  increased  by  4  equals  one  sixth 
of  the  second,  and  the  second  diminished  by  6  is  eight  times  the  fii-st ; 
what  are  the  numbers  ? 

132.  What  is  the  value  in  Federal  money  of  13  shares  of  Bank  stock, 
par  value  $125  per  share,  and  sold  at  11^  per  cent,  advance  ? 

133.  What  is  the  square  root  of  14734? 

134.  What  is  the  interest  of  $1974.36,  for  3  yrs.  2  mo.  17  days,  at  5| 
per  cent.  ? 

135.  What  is  the  14th  term  of  a  geometrical  series  the  first  term  of 
which  is  4  and  the  ratio  |  ? 

136.  What  is  the  16th  term  of  a  geometrical  series  the  first  tenn  of 
which  is  7  and  the  ratio  1|  ? 

137.  A  sells  to  B  5  loads  of  wood,  measuring,  1st,  8  ft.  6  in..  4  ft.,  3  ft. 
9  in.;  2d,  9  ft.,  4  ft.  2  in.,  3  ft.  11  in.;  3d,  9  ft,  2  in.,  4  ft.  4  in.,  4  ft.  1  in.; 
4th,  8  ft.  2  in.,  3  ft.  7  in.,  4  ft.  2  in. ;  5th,  7  ft.  11  in.,  4  ft.,  3  ft.  6  in ;  at 
$4.75  per  cord.  He  receives  in  payment,  47  bushels  of  oats,  at  38  cents 
per  busliel ;  56  lbs.  cheese,  at  8^  cents  per  lb. ;  and  the  balance  in  butter, 
at  15|  cents  per  lb. ;  how  much  butter  does  he  receive  ? 

138.  What  is  the  weight  of  one  rod  of  lead  pipe  one  fourth  of  an  inch 
in  thickness,  if  the  inner  diameter  measures  l^-  inches  ? 

139.  What  is  the  weight  of  a  plate  of  iron  half  an  inch  in  thickness, 
4  feet  long,  and  2  feet  3  inches  in  breadth '{ 

140.  How  many  feet  of  silver  wire,  one  tenth  of  an  inch  in  diameter, 
can  be  made  from  one  pound  avoirdupois  of  silver  1 

141.  How  many  gallons,  imperial  measure,  will  a  cylindrical  cistern 
hold,  3  feet  in  diameter  and  4^  feet  deep  ? 

142.  What  is  the  cost  of  transporting  64  barrels  of  flour,  each  contain- 
ing 7  qrs.  gross,  100  miles,  at  $3.12|^  per  ton,  allowing  for  the  weight  of 
each  cask  16  lbs.? 

143.  If  freight  by  rail  road  is  $3.12^  per  ton,  for  one  hundred  miles,  and 
freight  by  wagon  road  is  $20  dollars  per  ton  for  80  miles,  how  much  is 
saved  in  the  freight  of  a  barrel  of  flour  100  miles  by  rail  road,  allowing 
its  weight  to  be  as  in  the  preceding  example  ? 

144.  At  the  rate  named  above,  how  far  could  a  barrel  of  flour  be  car- 
ried by  wagon  road,  before  the  freight  should  amount  to  as  much  as  the 
flour  was  worth,  when  the  price  is  $5.62  per  barrel  ? 

145.  If  the  distance  from  New  York  to  Liverpool  is  3000  miles,  what 
would  be  the  cost  of  transporting  a  barrel  of  flour  that  distance,  at  the 
rate  of  $20  per  ton  for  every  80  miles  ? 

146.  If  a  barrel  of  flour  can  be  transported  from  New  York  to  Liver- 
pool for  65  cents,  what  would  that  give  for  the  transport  of  one  ton  80 
miles  ? 

147.  The  summit  of  the  Rocky  mountains,  visited  by  Freemont,  is  in 
Longitude  110  8';  what  time  is  it  at  Greenwich  when  it  is  noon  there  ? 

148.  There  is  a  field  20  rods  long  and  8  rods  broad,  with  a  path  3^  feet 
wide  running  round  it;  how  many  square  feet  are  there  in  the  path  ? 

149.  What  is  the  cost  of  excavating  a  cubical  pit,  measuring  7^  feet  in 
each  direction,  at  31^  cents  per  cubic  yardl 


APPENDIX.  221 

150.  What  is  the  Aveight  of  an  iron  cannon  9^  feet  in  length,  26  inches 
in  diameter  at  the  larger  end,  and  18  inches  at  the  smaller,  with  a  bore 
9  feet  long  and  10  inches  in  diameter,  allowing  for  no  inequalities  in  the 
surface  ? 

151.  What  will  be  the  duties  on  an  invoice  of  goods  amounting  to 
$1 156.80,  at  30  per  cent.  1  • 

152.  Bought  an  invoice  of  imported  goods,  amounting  to  £564  153. 
I  agree  to  give  18  per  cent,  in  advance  of  the  invoiced  price;  what  is  the 
amount  paid  in  Federal  money,  reckoning  $4,444  to  the  pound  ? 

153.  I  buy  for  cash  an  invoice  of  imported  goods,  amounting  to  £1146 
16s.;  what  is  the  invoiced  price  in  Federal  money,  allowing  Sterling 
money  to  be  9  per  cent,  in  advance  of  the  nominal  par  value? 

154.  A  merchant  owes  $16472.50;  he  fails,  his  means  of  payment  amount- 
ing to  only  $4345.62 ;  how  much  is  a  creditor  entitled  to,  who  holds  a 
note  against  him  of  $100,  dated  7^  months  previous  to  the  final  settle* 
ment,  and  promising  interest  60  days  after  date  1 

155.  A  road  three  and  a  half  rods  wide  is  laid  out  1  mile  and  14^  rods 
in  length,  for  which  damages  are  awarded  to  the  land-owners,  as  follows : 
for  80  rods  of  the  road,  at  the  rate  of  37  dollars  per  acre ;  for  110  rods,  at 
the  rate  of  22  dollais  per  acre;  and  for  the  remainder,  at  the  rate  of  30^ 
dollars  per  acre  ;  what  is  the  whole  amount  of  the  damages  1 

156.  What  is  due  for  the  freight  of  12  barrels  of  flour  65  miles,  at  f  of 
a  cent  per  lb.,  allowing  each  barrel  to  contain  7  qrs.  gross,  of  flour,  and 
the  cask  to  weigh  18  lbs.  ? 

157.  If  5  barrels  of  flour  suffice  for  a  family  of  11  persons  7  months, 
how  many  barrels  will  suffice  for  a  family  of  15  persons  4^  months  1 

158.  13^  times  144  is  7^  times  what  number? 

7i  -  5 

159.  384  is  —  of  how  many  times  1^  • 

160.  Reduce  62^  cents  to  the  decimal  of  a  £,  at  nominal  par  value. 

161.  How  many  seconds  were  there  in  the  year  18441 

162.  The  report  of  a  signal  gun  fired  on  the  equator,  at  12  o'clock,  is 
heard  at  a  place  due  west,  distant  16  miles ;  at  what  time  is  the  report  heard 
at  the  latter  place,  allowing  69^  miles  to  a  degree  of  longitude,  and  sound 
to  move  1  mile  and  10  rods  in  5  seconds  1 

163.  A  communication  is  made  by  the  magnetic  telegraph  from  Boston 
to  Washington,  at  1  o'clock,  P.  M. ;  at  what  time  will  it  be  received  at 
Washington,  allowing  no  time  for  the  transmission  of  the  fluid ;  longi- 
tude of  Boston  being  71°  4'  9";  that  of  Washington.  77°  1'  24"  ? 

164.  A  engaging  in  partnership  with  B  and  C,  puts  in  1600  dollars,  for 
9^  months:  B  puts  in  3100  dollars,  for  14  months;  C  puts  in  2200  dol- 
lars, for  12^  months.  They  gain  1046  dollars;  what  is  each  one's  share 
of  the  gain? 

165.  If  100  dollars  in  one  year  gain  6^  dollars  interest,  what  will  467 
dollars  gain  in  9^  months,  at  the  same  rate  1 

166.  What  is  the  value  of  17  shares  of  Bank  stock,  par  value  60  dollars 
per  share,  and  sold  at  6^^  per  cent,  advance  ? 

167.  A  man  buys  640  barrels  of  flour  at  $5|  per  bbl.;  pays  for  freight, 
$42.75;  for  storage,  $11.17;  and  sells  it  for  6^^.  per  baiTcl,  allowing  a 
commission  of  2^  per  cent. ;  what  was  his  loss  or  gain  per  cent.  ? 

168.  The  pipe  of  an  aqueduct,  12  inches  in  diameter,  is  divided  into 
two  branches,  such  that  thek  united  capacity  is  equal  to  that  of  the  main 

19* 


222  APPENDIX. 

pipe,  and  the  diameter  of  one  of  the  branches  is  9  inches ;  what  is  the 
diameter  of  the  other? 

169.  The  pipe  of  an  aqueduct  is  3  feet  in  diameter;  what  number  of 
pipes,  each  2^  inches  in  diameter,  would  have  a  capacity  equal  to  that  of 
the  main  pipe  1 

17u.  If  a  tree  measures  at  the  distance  of  2  feet  from  the  ground  12 
feet  in  circumference,  and  at  14  feet  from  the  ground  divides  into  two 
branches,  measuring  respectively  9  feet  and  7  feet  in  circumference,  how 
many  square  inches  more  of  surface  would  the  lower  horizontal  section  of 
the  tree  contain  than  the  upper  one  ? 

171.  A  certain  tree  measures  at  the  distance  of  3  feet  from  the  ground 
22^  feet  in  circumference  ;  at  some  distance  above,  its  four  branches  meas- 
ure respectively,  8  ft.  4  in.,  9  ft.,  7  ft.  6  in.,  and  6  ft.  5  in.  in  circumference ; 
what  is  the  ratio  of  the  magnitude  of  the  tree  at  the  lower,  compared  with 
its  magnitude  at  the  higher  place  of  measurement  ? 

172.  The  shadow  of  a  certain  tree,  as  cast  upon  the  ground,  measures 
102  feet  in  diameter;  allowing  the  shadow  to  be  circular,  how  many  rods 
of  ground  does  it  cover  ? 

173.  How  many  cubic  inches  does  a  wine  glass  contain,  measuring  3 J 
inches  in  depth,  and  2  inches  in  diameter  at  the  top,  the  form  being  that 
of  an  inverted  cone  ? 

174.  How  many  yards  of  lining,  f  of  a  yard  wide,  will  line  37^  yards 
of  cloth  1^   yards  wide  1 

175.  What  is  the  3d  term  of  the  square  900-f-4204-n  ? 

176.  What  is  the  3d  term  of  the  square  1600+480-f-n  ? 

177.  Complete  the  square  4900-f-n-f-64. 

178.  What  is  the  4th  term  of  the  cube  27000+5400+360+0  ? 

179.  What  is  the  4th  term  of  the  cube  64000+4800+1 20+ Q  ? 

180.  What  is  the  4th  term  of  th^  cube  125000+22500+1350+C  ? 

181.  What  is  the  cube  root  of  68437 1 

182.  What  is  the  cube  foot  of  954326 1 

183.  What  is  the  solid  contents  of  the  largest  sphere  that  can  be  cut 
from  a  cubic  block  13|  inches  on  a  side  1 

184.  From  a  sphere  measuring  10  inches  in  diameter,  the  largest  pos- 
sible cubic  block  has  been  cut ;  and  from  this  block  again  the  largest  pos- 
sible sphere  has  been  cut ;  what  is  the  diameter  of  the  last  named  sphere  1 

185.  From  a  sphere  20  inches  in  diameter  what  are  the  dimensions  of 
the  largest  parallel  prism  that  can  be  cut,  making  the  length  to  the 
breadth  as  4  to  3,  and  the  breadth  to  the  thickness  as  3  to  2  ? 

186.  How  many  cubic  feet  are  there  in  the  walls  of  a  brick  house,  the 
length  and  breadth  of  which  are  each  44  feet  outside,  and  the  height  24 
feet,  supposing  the  walls  to  be  perpendicular  outside,  and  the  thickness  to 
be  2  feet  at  the  bottom,  and  1  foot  at  the  top,  making  no  deduction  for 
doors  or  windows  1 

187.  What  are  the  prime  factors  of  3746  ?     of  9862  ? 

188.  What  is  the  least  common  multiple  of  684,  963,  and  8416 1 

189.  What  is  the  greatest  common  divisor  of  94620,  3642,  and  1646  ? 

190.  How  many  times  will  the  wheel  of  a  rail  road  car,  if  it  be  2^  feet 
in  diameter,  revolve  in  going  40  miles  1 

191.  How  many  times  will  such  a  wheel  revolve  in  a  minute,  if  the 
speed  of  the  car  be  20  miles  an  hour  1 

192.  What  is  the  cost  in  Federal  money  of  the  freight  of  940  bales  of 
cotton,  averaging  340  lbs.  per  bale,  at  |  of  a  penny  per  lb,  ? 


ArpEN'Dix.  223 

193.  Sold  34  pieces  cotton  goods,  averaging  31|-  yards  each  in  length, 
at  9|  cents  per  yard,  1^  per  cent,  off;  what  is  the  amount  of  the  bill  1 

194.  Bought  840  barrels  of  flour  on  six  months'  credit,  at  4|  dollars  per 
ban-el ;  sold  the  flour  the  same  day  on  three  months'  credit,  at  4i^  dollars 
per  barrel ;  did  I  gain  or  lose,  and  how  much,  estimating  the  present 
worth  of  the  debts  at  the  date  of  the  transaction  7 

195.  A  merchant  buys  for  me,  on  commission,  400  barrels  of  flour,  for 
$4.34  per  barrel,  cash ;  he  sells  the  flour  on  the  same  day  for  cash,  at  $4.46 
per  barrel ;  how  much  do  I  gain  by  the  operation,  allowing  1^  per  cent, 
commission  on  the  i)urchases,  and  2  per  cent,  on  the  sales  ? 

196.  A  sets  out  on  a  journey,  travelling  24  miles  a  day;  B  sets  out  2 
days  after,  travelling  31§  miles  a  day;  A,  after  travelling  3  days,  goes  25 
miles  a  day ;  and  B,  after  travelling  4  days,  goes  33^  miles  a  day ;  in  how 
many  days  after  A  sets  out,  will  B  overtake  him  ? 

197.  If  a  parallel  prism  is  4^  inches  thick,  5  inches  wide  and  6  inches 
long,  what  must  be  the  diameter  of  the  hollow  sphere  that  would  enclose  it  ? 

198.  If  18  horses  in  16  weeks  consume  204  bushels  of  oats,  how  many 
horses  will  it  require  to  consume  413  bushels  in  18  weeks  ? 

199.  If  100  dollars  gain  6  dollars  interest  in  12  months,  what  will  be 
the  interest  of  840  dollars  for  5^^  months  ? 

200.  If  a  field  containing  17  acres  measures  81  rods  on  one  side,  what 
must  be  the  length  of  the  corresponding  side  of  a  similar  field  containing 
26^  acres  '?  JT 

201.  The  contents  of  two  similar  fields  fft-e  as  4^  to  7,  and  the  smaller 
measures  on  one  side  63  rods :  what  must;be  the  corresponding  dimen- 
sion of  the  larger  field  ? 

202.  There  are  two  circles  ;  their  areas  are  as  14  to  19,  and  the  diame- 
ter of  the  smaller  is  16  rods  ;  what  is  the  diameter  of  the  larger  ? 

'203.  What  is  the  area  of  a  circle  ■v\^j||pediameter'is  44  feet  ? 

204.  "What  is  the  circumference  of  J^im^\^j|P^ diameter  is  67  inches? 

205.  Given  the  circumference  of  a  circle  60  rods  to  find  its  area. 

206.  There  are  three  equilateral  triangles  whose  areas  are  to  each  oth- 
er as  the  numbers  3,  4,  and  7 ;  a  side  of  the  smallest  measures  40  rods; 
what  is  the  sum  of  their  areas  ? 

207.  A  grindstone  is  3  feet  in  diameter ;  allowing  the  hole  in  the  mid- 
dle to  be  2  inches  in  diameter,  how  many  inches  must  be  ground  off  to 
grind  away  half  of  the  stone  ? 

208.  The  fore  wheels  of  a  waggon  are  3  feet  10  inches  in  diameter,  the 
hind  wheels  4  feet  2  inches  in  diameter,  how  many  times  more  does  one 
of  the  fore  wheels  turn  round  than  one  of  the  hind  wheels  in  going  one 
mile? 

209.  What  is  the  weight  of  a  cast  iron  cylinder  6  feet  long  and  4^ 
inches  in  diameter,  the  specific  gravity  being  as  already  given  1 

210.  The  frustum  of  a  cone  7  feet  long  is  14  inches  in  diameter  at  the 
larger  end,  and  10  inches  in  diameter  at  the  smaller ;  how  faf  from  the 
base  must  it  be  cut  in  two  to  divide  it  into  equal  parts  ? 

211.  What  is  the  first  prime  number  above  901  ? 

212.  What  is  the  first  prime  number  below  10000? 

213.  What  is  the  greatest  common  divisor  of  1846,  3105,  684  and  1006  ? 

214.  What  are  all  the  prime  factors  of  801,  of  3042,  of  586,  of  908  ? 

215.  In  what  proportion  may  corn  at  80  cents,  be  mixed  with  rye  at 
86  cents,  and  with  oats  at  43  cents  per  bushel,  to  make  the  mixture  worth 
50  cents  per  bushel  ? 


224  .  Kt'E!>L>JX. 

S-l       4.9         1 Q 

216.  Add  the  fractions  ^4-^+-if . 

7        61:5^      S/g- 

217.  What  is  the  value  of  ^  of  ^  £  f  of  ^tj  S,  expressed  in  the  de- 
cimal of  a  £  ? 

218.  What  is  the  present  value  of  a  note  of  584  dollars,  payable  in 
three  months  ? 

219.  What  is  the  bank  discount  on  a  note  of  150  dollars,  payable  in 
three  months  1 

220.  What  sum  will  be  paid  on  a  note  of  240  dollars,  discounted  at  a 
bank,  for  90  days  1 

221.  What  is  the  interest  of  1200  dollars  for  10  days,  at  7  per  cent.  ? 

222.  Divide  the  sum  of  the  decimals  2016  +  9172  +  0064,  by  f  of  f 
reduced  to  a  decimal. 

223.  Divide  3  T.  17  cwt.  3  qrs.  19  lbs.  by  6. 

224.  Divide  9  m.  3  fur.  21  r.  14  ft.  bv  8. 

225.  Multiply  31  d.  14  h.  37  m.  15  sec.  by  19. 

226.  Multiply  83  A.  3  R.  22  r.  by  12. 

"227.  What  is  one  fifth  of  16  T.  11  cwt.  2  qrs.  20  lbs.  ? 

228.  What  is  the  value  at  4  dollars  a  cord,  of  three  loads  of  wood 
measuring  as  follows :  1st,  8  ft.  6  in.,  4  ft  2  in.,  3  ft.  9  in. ;  2d,  9  ft.,  4  ft. 
1  in.,  3  ft.  10  in. ;  3d,  8  ft.  1  in.,  4  ft.,  4  ft.  2  in.  ? 

229.  What  is  the  15th  term  of  an  arithmetical  series,  the  first  term  of 
which  is  3,  and  the  commSflKifFerence  ^  ? 

230.  If  the  9th  term  of  a^  arithmetical  series  is  23,  and  the  common 
diiference  §,  ^vhat  is  the  3d  ^rm  ? 

231.  What  is  the  sum  of  an  arithmetical  series  of  40  terms,  if  the  first 
term  is  2  and  the  common  difference  31.  ? 

232.  What  is  thersum  of  an  aathmetical  series  of  72  terms,  if  the  first 
term  is  1  and  the  conmon'jM^TOce  1^  ? 

233.  A  man  engageS  to  walk  1000  miles  in  1000  hours,  on  condition 
of  receiving  1  cent  for  the  first  mile,  and  for  each  mile  after,  5-  of  a  cent 
more  than  he  had  for  the  mile  preceding  it;  what  will  he  be  entitled  to 
on  the  fulfillment  of  his  contract? 

234.  There  are  two  grindstones  the  thickness  of  which  is  to  the  diam- 
eter as  2  to  11 ;  the  smaller  one  is  2  feet  in  diameter,  the  other  is  three 
times  as  heavy  ;  what  is  its  diameter  ? 

235.  A  man  has  a  triangular  field  containing  7  acres,  tlie  vertex  or 
point  of  the  triangle  is  54  rods  from  the  base,  at  what  distance  from  the 
base  must  a  line  be  drawn  parallel  to  it  so  as  to  cut  off  one  half  the  Held  ? 

236.  The  hypotenuse  and  perpendicular  of  a  triangle  measure  togeth- 
er 816  rods,  the  base  measures  61  rods ;  what  is  the  length  of  the  ])erpen- 
dicular  1 

237.  A  rope  100  feet  long  passes  straight  from  the  ground  at  flic  dis- 
tance of  10  feet  from  a  perpendicular  pole,  over  the  top  of  the  polo,  which 
is  25  feet  in  height,  and  thence  is  drawn  so  as  to  reach  the  ground  at  tli<j 
farthest  possible  point ;  allowing  the  ground  to  be  level,  how  fiir  is  the 


last  named  point  from  the  foot  of  the  pol 

238.  A  stick' of  timber  in  the  form  of  a  truncated  wedge  is  10  feet 
long,  2  feet  wide  through  its  whole  extent,  20  inches  in  thickness  at  one 
end,  and  14  inches  thick  at  the  other;  how  far  from  the  thicker  end  must 
it  be  cut  in  two  so  as  to  divide  it  into  two  equal  parts  ? 


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